Daftar masalah matematika yang belum terpecahkan: Perbedaan antara revisi
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* Mencari |
* Mencari pemadanan batas atas dan bawah untuk [[Himpunan-k (geometri)|himpunan-''k'']] dan membagi garis<ref>{{citation |
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| last = Dey | first = Tamal K. | author-link = Tamal Dey |
| last = Dey | first = Tamal K. | author-link = Tamal Dey |
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| doi = 10.1007/PL00009354 |
| doi = 10.1007/PL00009354 |
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=== [[Teori graf]] === |
=== [[Teori graf]] === |
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==== |
==== Lintasan dan siklus dalam graf ==== |
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* [[ |
* [[Konjektur Barnette]] bahwa setiap graf planar tiga terhubung dwipihak kubik memiliki sebuah siklus Hamilton<ref>{{citation |
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| last = Florek | first = Jan |
| last = Florek | first = Jan |
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| doi = 10.1016/j.disc.2010.01.018 |
| doi = 10.1016/j.disc.2010.01.018 |
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| volume = 310 |
| volume = 310 |
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| year = 2010}}.</ref> |
| year = 2010}}.</ref> |
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* [[Kekerasan graf|Konjektur kekerasan Chvátal]], bahwa terdapat sebuah bilangan <math>t</math> sehingga setiap graf keras-<math>t</math> adalah Hamilton<ref>{{citation |
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* [[Graph toughness|Chvátal's toughness conjecture]], that there is a number {{mvar|t}} such that every {{mvar|t}}-tough graph is Hamiltonian<ref>{{citation |
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| last1 = Broersma | first1 = Hajo |
| last1 = Broersma | first1 = Hajo |
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| last2 = Patel | first2 = Viresh |
| last2 = Patel | first2 = Viresh |
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| volume = 75 |
| volume = 75 |
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| year = 2014}}</ref> |
| year = 2014}}</ref> |
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* |
* [[Konjektur peliputan ganda siklus]] bahwa setiap yang tak memiliki jembatan, memiliki sebuah keluarga siklus yang termasuk setiap tepi dua kali<ref>{{citation |
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| last = Jaeger | first = F. |
| last = Jaeger | first = F. |
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| contribution = A survey of the cycle double cover conjecture |
| contribution = A survey of the cycle double cover conjecture |
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Baris 385: | Baris 385: | ||
| year = 1985| isbn = 9780444878038 |
| year = 1985| isbn = 9780444878038 |
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}}.</ref> |
}}.</ref> |
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* |
* [[Konjektur Erdős–Gyárfás]] pada siklus dengan panjang pangkat dari dua dalam graf kubik<ref>{{citation|title=Erdös-Gyárfás conjecture for cubic planar graphs|first1=Christopher Carl|last1=Heckman|first2=Roi|last2=Krakovski|volume=20|issue=2|year=2013|at=P7|journal=Electronic Journal of Combinatorics|doi-access=free|doi=10.37236/3252}}.</ref> |
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* |
* Konjektur [[arborisitas linear]] pada penguraian graf menjadi gabungan lepas lintasan menurut derajat maksimumnya<ref>{{citation |
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| last1 = Akiyama | first1 = Jin | author1-link = Jin Akiyama |
| last1 = Akiyama | first1 = Jin | author1-link = Jin Akiyama |
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| last2 = Exoo | first2 = Geoffrey |
| last2 = Exoo | first2 = Geoffrey |
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Baris 398: | Baris 398: | ||
| volume = 11 |
| volume = 11 |
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| year = 1981}}.</ref> |
| year = 1981}}.</ref> |
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* |
* [[Konjektur Lovász]] pada lintasan Hamilton dalam graf simetrik<ref>[[László Babai|L. Babai]], [http://www.cs.uchicago.edu/research/publications/techreports/TR-94-10 Automorphism groups, isomorphism, reconstruction] {{Webarchive|url=https://web.archive.org/web/20070613201449/http://www.cs.uchicago.edu/research/publications/techreports/TR-94-10 |date=2007-06-13 }}, in ''Handbook of Combinatorics'', Vol. 2, Elsevier, 1996, 1447–1540.</ref> |
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* |
* [[Masalah Oberwolfach]] di mana 2 graf beraturan memilik sifat bahwa sebuah graf sempurna pada jumlah puncak yang sama dapat diuraikan menjadi salinan tepi-lepas dari graf yang diberikan.<ref>{{citation |
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| last1 = Lenz | first1 = Hanfried |
| last1 = Lenz | first1 = Hanfried |
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| last2 = Ringel | first2 = Gerhard |
| last2 = Ringel | first2 = Gerhard |
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Baris 410: | Baris 410: | ||
| volume = 97 |
| volume = 97 |
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| year = 1991}}</ref> |
| year = 1991}}</ref> |
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* [[ |
* [[Konjektur Szymanski]] |
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==== |
==== Pewarnaan and pelabelan graf ==== |
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[[Image:Erdős–Faber–Lovász conjecture.svg|thumb|upright=1.2|An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.]] |
[[Image:Erdős–Faber–Lovász conjecture.svg|thumb|upright=1.2|An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.]] |
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* [[Cereceda |
* [[Konjektur Cereceda]] pada diameter dari ruang pewarnaan graf merosot<ref>{{citation |
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| last1 = Bousquet | first1 = Nicolas |
| last1 = Bousquet | first1 = Nicolas |
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| last2 = Bartier | first2 = Valentin |
| last2 = Bartier | first2 = Valentin |
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Baris 429: | Baris 429: | ||
| year = 2019| s2cid = 195791634 |
| year = 2019| s2cid = 195791634 |
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}}</ref> |
}}</ref> |
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* |
* [[Konjektur Erdős–Faber–Lovász]] pada gabungan pewarnaan klik<ref>{{citation |
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| last1 = Chung | first1 = Fan | author-link1 = Fan Chung |
| last1 = Chung | first1 = Fan | author-link1 = Fan Chung |
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| last2 = Graham | first2 = Ron | author-link2 = Ronald Graham |
| last2 = Graham | first2 = Ron | author-link2 = Ronald Graham |
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Baris 436: | Baris 436: | ||
| publisher = A K Peters |
| publisher = A K Peters |
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| pages = 97–99}}.</ref> |
| pages = 97–99}}.</ref> |
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* [[Konjektur Gyárfás–Sumner]] pada keterbatasan <math>\chi</math> dari graf dengan sebuah pohon terimbas yang dliarang<ref>{{citation |
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* The [[Gyárfás–Sumner conjecture]] on χ-boundedness of graphs with a forbidden induced tree<ref>{{citation |
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| last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky |
| last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky |
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| last2 = Seymour | first2 = Paul | author2-link = Paul Seymour (mathematician) |
| last2 = Seymour | first2 = Paul | author2-link = Paul Seymour (mathematician) |
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Baris 448: | Baris 448: | ||
| year = 2014| doi-access = free |
| year = 2014| doi-access = free |
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}}</ref> |
}}</ref> |
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* |
* [[Konjektur Hadwiger (teori graf)|Konjektur Hadwiger]] mengaitkan pewarnaan untuk minor klik<ref>{{citation |
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| last = Toft | first = Bjarne |
| last = Toft | first = Bjarne |
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| journal = Congressus Numerantium |
| journal = Congressus Numerantium |
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Baris 456: | Baris 456: | ||
| volume = 115 |
| volume = 115 |
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| year = 1996}}.</ref> |
| year = 1996}}.</ref> |
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* |
* [[Masalah Hadwiger–Nelson]] pada bilangan kromatik dari graf jarak satuan<ref>{{citation |
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| last1 = Croft | first1 = Hallard T. |
| last1 = Croft | first1 = Hallard T. |
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| last2 = Falconer | first2 = Kenneth J. |
| last2 = Falconer | first2 = Kenneth J. |
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Baris 463: | Baris 463: | ||
| publisher = Springer-Verlag |
| publisher = Springer-Verlag |
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| year = 1991}}, Problem G10.</ref> |
| year = 1991}}, Problem G10.</ref> |
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* [[ |
* [[Konjektur pewarnaan Jaeger's Petersen]] bahwa setiap grafik kubik takberjembantan memiliki sebuah pemetaan siklus-kontinu ke graf Petersen<ref>{{citation |
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| last1 = Hägglund |
| last1 = Hägglund |
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| first1 = Jonas |
| first1 = Jonas |
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| doi-access = free |
| doi-access = free |
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}}.</ref> |
}}.</ref> |
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* |
* [[Daftar pewarnaan konjektur]] bahwa, untuk setiap grad, daftar kromatik indeks sama dengan indeks kromatik<ref>{{citation|last1=Jensen|first1=Tommy R.|last2=Toft|first2=Bjarne|year=1995|title=Graph Coloring Problems|location=New York|publisher=Wiley-Interscience|isbn=978-0-471-02865-9|chapter=12.20 List-Edge-Chromatic Numbers|pages=201–202}}.</ref> |
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* |
* [[Konjektur pewarnaan total]] Behzad dan Vizing bahwa bilangan kromatik total paling banyak dua ditambah derajat maksimum<ref>{{citation |
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| last1 = Molloy | first1 = Michael |
| last1 = Molloy | first1 = Michael |
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| last2 = Reed | first2 = Bruce | author1-link = Bruce Reed (mathematician) |
| last2 = Reed | first2 = Bruce | author1-link = Bruce Reed (mathematician) |
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* [[Turán's brick factory problem]] – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?<ref>{{citation | last1 = Pach | first1 = János | author1-link = János Pach | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir | contribution = 5.1 Crossings—the Brick Factory Problem | pages = 126–127 | publisher = [[American Mathematical Society]] | series = Mathematical Surveys and Monographs | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures | volume = 152 | year = 2009}}.</ref> |
* [[Turán's brick factory problem]] – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?<ref>{{citation | last1 = Pach | first1 = János | author1-link = János Pach | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir | contribution = 5.1 Crossings—the Brick Factory Problem | pages = 126–127 | publisher = [[American Mathematical Society]] | series = Mathematical Surveys and Monographs | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures | volume = 152 | year = 2009}}.</ref> |
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* [[Universal point set]]s of subquadratic size for planar graphs<ref>{{citation | last1 = Demaine | first1 = E. | author1-link = Erik Demaine | last2 = O'Rourke | first2 = J. | author2-link = Joseph O'Rourke (professor) | contribution = Problem 45: Smallest Universal Set of Points for Planar Graphs | title = The Open Problems Project | url = http://cs.smith.edu/~orourke/TOPP/P45.html | year = 2002–2012 | access-date = 2013-03-19 | archive-url = https://web.archive.org/web/20120814154255/http://cs.smith.edu/~orourke/TOPP/P45.html | archive-date = 2012-08-14 | url-status = live }}.</ref> |
* [[Universal point set]]s of subquadratic size for planar graphs<ref>{{citation | last1 = Demaine | first1 = E. | author1-link = Erik Demaine | last2 = O'Rourke | first2 = J. | author2-link = Joseph O'Rourke (professor) | contribution = Problem 45: Smallest Universal Set of Points for Planar Graphs | title = The Open Problems Project | url = http://cs.smith.edu/~orourke/TOPP/P45.html | year = 2002–2012 | access-date = 2013-03-19 | archive-url = https://web.archive.org/web/20120814154255/http://cs.smith.edu/~orourke/TOPP/P45.html | archive-date = 2012-08-14 | url-status = live }}.</ref> |
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==== Word-representation of graphs ==== |
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*Characterise (non-)[[Word-representable graph|word-representable]] [[planar graph]]s <ref name="KL15">[https://www.springer.com/la/book/9783319258577 S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015.]</ref><ref name="K17">[[arxiv:1705.05924|S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67.]] </ref><ref name="KP18">[https://link.springer.com/article/10.1134/S1990478918020084 S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296.]</ref><ref name="KP18-2">[http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=da&paperid=894&option_lang=rus С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53]</ref> |
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*Characterise [[Word-representable graph|word-representable]] near-triangulations containing the complete graph ''K''<sub>4</sub> (such a characterisation is known for ''K''<sub>4</sub>-free planar graphs <ref name="Glen2019">{{cite arxiv |eprint=1605.01688|author1=Marc Elliot Glen|title=Colourability and word-representability of near-triangulations|class=math.CO|year=2016}}</ref>) |
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*Classify graphs with representation number 3, that is, graphs that can be [[Word-representable graph|represented]] using 3 copies of each letter, but cannot be represented using 2 copies of each letter <ref name="Kit2013-3-repr">[[arxiv:1403.1616|S. Kitaev. On graphs with representation number 3, J. Autom., Lang. and Combin. 18 (2013), 97−112.]]</ref> |
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*Is the [[line graph]] of a non-[[Word-representable graph|word-representable]] graph always non-[[Word-representable graph|word-representable]]? <ref name="KL15">[https://www.springer.com/la/book/9783319258577 S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015.]</ref><ref name="K17">[[arxiv:1705.05924|S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67.]] </ref><ref name="KP18">[https://link.springer.com/article/10.1134/S1990478918020084 S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296.]</ref><ref name="KP18-2">[http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=da&paperid=894&option_lang=rus С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53]</ref> |
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*Are there any graphs on ''n'' vertices whose [[Word-representable graph|representation]] requires more than floor(''n''/2) copies of each letter? <ref name="KL15">[https://www.springer.com/la/book/9783319258577 S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015.]</ref><ref name="K17">[[arxiv:1705.05924|S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67.]] </ref><ref name="KP18">[https://link.springer.com/article/10.1134/S1990478918020084 S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296.]</ref><ref name="KP18-2">[http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=da&paperid=894&option_lang=rus С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53]</ref> |
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*Is it true that out of all [[bipartite graph]]s [[crown graph]]s require longest word-representants? <ref name="GKP18">{{cite journal|url = https://www.sciencedirect.com/science/article/pii/S0166218X18301045 | doi=10.1016/j.dam.2018.03.013 | volume=244 | title=On the representation number of a crown graph | year=2018 | journal=Discrete Applied Mathematics | pages=89–93 | last1 = Glen | first1 = Marc | last2 = Kitaev | first2 = Sergey | last3 = Pyatkin | first3 = Artem| s2cid=46925617 }}</ref> |
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*Characterise [[word-representable graph]]s in terms of (induced) forbidden subgraphs. <ref name="KL15">[https://www.springer.com/la/book/9783319258577 S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015.]</ref><ref name="K17">[[arxiv:1705.05924|S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67.]] </ref><ref name="KP18">[https://link.springer.com/article/10.1134/S1990478918020084 S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296.]</ref><ref name="KP18-2">[http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=da&paperid=894&option_lang=rus С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53]</ref> |
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*Which (hard) problems on graphs can be translated to words [[Word-representable graph|representing]] them and solved on words (efficiently)? <ref name="KL15">[https://www.springer.com/la/book/9783319258577 S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015.]</ref><ref name="K17">[[arxiv:1705.05924|S. Kitaev. A comprehensive introduction to the theory of word-representable graphs. In: É. Charlier, J. Leroy, M. Rigo (eds), Developments in Language Theory. DLT 2017. Lecture Notes Comp. Sci. 10396, Springer, 36−67.]] </ref><ref name="KP18">[https://link.springer.com/article/10.1134/S1990478918020084 S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296.]</ref><ref name="KP18-2">[http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=da&paperid=894&option_lang=rus С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53]</ref> |
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==== Miscellaneous graph theory ==== |
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* [[Conway's 99-graph problem]]: does there exist a [[strongly regular graph]] with parameters (99,14,1,2)?<ref>{{citation |
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| last = Conway |
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| first = John H. |
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| author-link = John Horton Conway |
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| access-date = 2019-02-12 |
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| publisher = Online Encyclopedia of Integer Sequences |
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| title = Five $1,000 Problems (Update 2017) |
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| url = https://oeis.org/A248380/a248380.pdf |
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| archive-url = https://web.archive.org/web/20190213123825/https://oeis.org/A248380/a248380.pdf |
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| archive-date = 2019-02-13 |
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| url-status = live |
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}}</ref> |
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* The [[Erdős–Hajnal conjecture]] on large cliques or independent sets in graphs with a forbidden induced subgraph<ref>{{citation |
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| last = Chudnovsky |
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| first = Maria |
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| author-link = Maria Chudnovsky |
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| arxiv = 1606.08827 |
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| doi = 10.1002/jgt.21730 |
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| issue = 2 |
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| journal = Journal of Graph Theory |
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| mr = 3150572 |
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| zbl = 1280.05086 |
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| pages = 178–190 |
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| title = The Erdös–Hajnal conjecture—a survey |
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| url = http://www.columbia.edu/~mc2775/EHsurvey.pdf |
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| volume = 75 |
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| year = 2014 |
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| s2cid = 985458 |
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| access-date = 2016-09-22 |
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| archive-url = https://web.archive.org/web/20160304102611/http://www.columbia.edu/~mc2775/EHsurvey.pdf |
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| archive-date = 2016-03-04 |
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| url-status = live |
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}}.</ref> |
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* The [[GNRS conjecture]] on whether minor-closed graph families have <math>\ell_1</math> embeddings with bounded distortion<ref>{{citation |
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| last1 = Gupta | first1 = Anupam |
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| last2 = Newman | first2 = Ilan |
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| last3 = Rabinovich | first3 = Yuri |
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| last4 = Sinclair | first4 = Alistair | author4-link = Alistair Sinclair |
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| doi = 10.1007/s00493-004-0015-x |
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| issue = 2 |
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| journal = [[Combinatorica]] |
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| mr = 2071334 |
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| pages = 233–269 |
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| title = Cuts, trees and <math>\ell_1</math>-embeddings of graphs |
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| volume = 24 |
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| year = 2004| citeseerx = 10.1.1.698.8978 |
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| s2cid = 46133408 |
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}}</ref> |
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* [[Graham's pebbling conjecture]] on the pebbling number of Cartesian products of graphs<ref>{{citation |
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| last = Pleanmani | first = Nopparat |
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| doi = 10.1142/s179383091950068x |
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| issue = 6 |
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| journal = Discrete Mathematics, Algorithms and Applications |
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| mr = 4044549 |
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| page = 1950068, 7 |
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| title = Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph |
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| volume = 11 |
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| year = 2019}}</ref> |
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* The [[implicit graph conjecture]] on the existence of implicit representations for slowly-growing [[Hereditary property#In graph theory|hereditary families of graphs]]<ref>{{citation|first=Jeremy P.|last=Spinrad|title=Efficient Graph Representations|year=2003|isbn=978-0-8218-2815-1|chapter=2. Implicit graph representation|pages=17–30|chapter-url=https://books.google.com/books?id=RrtXSKMAmWgC&pg=PA17}}.</ref> |
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* Jørgensen's conjecture that every 6-vertex-connected ''K''<sub>6</sub>-minor-free graph is an [[apex graph]]<ref>{{citation|url=http://www.openproblemgarden.org/op/jorgensens_conjecture|title=Jorgensen's Conjecture|work=Open Problem Garden|access-date=2016-11-13|archive-url=https://web.archive.org/web/20161114232136/http://www.openproblemgarden.org/op/jorgensens_conjecture|archive-date=2016-11-14|url-status=live}}.</ref> |
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* Meyniel's conjecture that [[cop number]] is <math>O(\sqrt n)</math><ref>{{citation |
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| last1 = Baird | first1 = William |
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| last2 = Bonato | first2 = Anthony |
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| arxiv = 1308.3385 |
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| doi = 10.4310/JOC.2012.v3.n2.a6 |
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| issue = 2 |
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| journal = Journal of Combinatorics |
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| mr = 2980752 |
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| pages = 225–238 |
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| title = Meyniel's conjecture on the cop number: a survey |
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| volume = 3 |
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| year = 2012| s2cid = 18942362 |
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}}</ref> |
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* Does a [[Moore graph]] with girth 5 and degree 57 exist?<ref>{{citation|last=Ducey|first=Joshua E.|doi=10.1016/j.disc.2016.10.001|issue=5|journal=[[Discrete Mathematics (journal)|Discrete Mathematics]]|mr=3612450|pages=1104–1109|title=On the critical group of the missing Moore graph|volume=340|year=2017|arxiv=1509.00327|s2cid=28297244}}</ref> |
|||
* What is the largest possible [[pathwidth]] of an {{mvar|n}}-vertex [[cubic graph]]?<ref>{{citation |
|||
| last1 = Fomin | first1 = Fedor V. |
|||
| last2 = Høie | first2 = Kjartan |
|||
| doi = 10.1016/j.ipl.2005.10.012 |
|||
| issue = 5 |
|||
| journal = Information Processing Letters |
|||
| mr = 2195217 |
|||
| pages = 191–196 |
|||
| title = Pathwidth of cubic graphs and exact algorithms |
|||
| volume = 97 |
|||
| year = 2006}}</ref> |
|||
* The [[reconstruction conjecture]] and [[new digraph reconstruction conjecture]] on whether a graph is uniquely determined by its vertex-deleted subgraphs.<ref>{{citation|first=Allen|last=Schwenk|title=Some History on the Reconstruction Conjecture|year=2012|url=http://faculty.nps.edu/rgera/conjectures/jmm2012/Schwenk,%20%20Some%20History%20on%20the%20RC.pdf|work=Joint Mathematics Meetings|access-date=2018-11-26|archive-url=https://web.archive.org/web/20150409233306/http://faculty.nps.edu/rgera/Conjectures/jmm2012/Schwenk,%20%20Some%20History%20on%20the%20RC.pdf|archive-date=2015-04-09|url-status=live}}</ref><ref>{{citation |
|||
| last = Ramachandran | first = S. |
|||
| doi = 10.1016/S0095-8956(81)80019-6 |
|||
| issue = 2 |
|||
| journal = Journal of Combinatorial Theory |
|||
| mr = 630977 |
|||
| pages = 143–149 |
|||
| series = Series B |
|||
| title = On a new digraph reconstruction conjecture |
|||
| volume = 31 |
|||
| year = 1981| doi-access = free |
|||
}}</ref> |
|||
* The [[second neighborhood problem]]: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?<ref>[https://faculty.math.illinois.edu/~west/openp/2ndnbhd.html Seymour's 2nd Neighborhood Conjecture] {{Webarchive|url=https://web.archive.org/web/20190111175310/https://faculty.math.illinois.edu/~west/openp/2ndnbhd.html |date=2019-01-11 }}, Open Problems in Graph Theory and Combinatorics, [[Douglas West (mathematician)|Douglas B. West]].</ref> |
|||
* Do there exist infinitely many [[strongly regular graph|strongly regular]] [[geodetic graph]]s, or any strongly regular geodetic graphs that are not Moore graphs?<ref>{{citation |
|||
| last1 = Blokhuis | first1 = A. |
|||
| last2 = Brouwer | first2 = A. E. | author-link = Andries Brouwer |
|||
| doi = 10.1007/BF00191941 |
|||
| issue = 1–3 |
|||
| journal = [[Geometriae Dedicata]] |
|||
| mr = 925851 |
|||
| pages = 527–533 |
|||
| title = Geodetic graphs of diameter two |
|||
| volume = 25 |
|||
| year = 1988| s2cid = 189890651 |
|||
}}</ref> |
|||
* [[Sumner's conjecture]]: does every <math>(2n-2)</math>-vertex tournament contain as a subgraph every <math>n</math>-vertex oriented tree?<ref>{{citation |
|||
| last1 = Kühn | first1 = Daniela | author1-link = Daniela Kühn |
|||
| last2 = Mycroft | first2 = Richard |
|||
| last3 = Osthus | first3 = Deryk |
|||
| arxiv = 1010.4430 |
|||
| doi = 10.1112/plms/pdq035 |
|||
| issue = 4 |
|||
| journal = Proceedings of the London Mathematical Society | series = Third Series |
|||
| mr = 2793448 | zbl=1218.05034 |
|||
| pages = 731–766 |
|||
| title = A proof of Sumner's universal tournament conjecture for large tournaments |
|||
| volume = 102 |
|||
| year = 2011| s2cid = 119169562 }}.</ref> |
|||
* Tutte's conjectures that every bridgeless graph has a [[nowhere-zero flows|nowhere-zero 5-flow]] and every [[Petersen graph|Petersen]]-[[Graph minor|minor]]-free bridgeless graph has a nowhere-zero 4-flow<ref>[http://www.openproblemgarden.org/op/4_flow_conjecture 4-flow conjecture] {{Webarchive|url=https://web.archive.org/web/20181126134908/http://www.openproblemgarden.org/op/4_flow_conjecture |date=2018-11-26 }} and [http://www.openproblemgarden.org/op/5_flow_conjecture 5-flow conjecture] {{Webarchive|url=https://web.archive.org/web/20181126134833/http://www.openproblemgarden.org/op/5_flow_conjecture |date=2018-11-26 }}, Open Problem Garden</ref> |
|||
* [[Vizing's conjecture]] on the [[domination number]] of [[cartesian product of graphs|cartesian products of graphs]]<ref>{{citation |
|||
| last1 = Brešar | first1 = Boštjan |
|||
| last2 = Dorbec | first2 = Paul |
|||
| last3 = Goddard | first3 = Wayne |
|||
| last4 = Hartnell | first4 = Bert L. |
|||
| last5 = Henning | first5 = Michael A. |
|||
| last6 = Klavžar | first6 = Sandi |
|||
| last7 = Rall | first7 = Douglas F. |
|||
| doi = 10.1002/jgt.20565 |
|||
| issue = 1 |
|||
| journal = Journal of Graph Theory |
|||
| mr = 2864622 |
|||
| pages = 46–76 |
|||
| title = Vizing's conjecture: a survey and recent results |
|||
| volume = 69 |
|||
| year = 2012| citeseerx = 10.1.1.159.7029 |
|||
}}.</ref> |
|||
* [[Zarankiewicz problem]] |
|||
=== [[Teori grup]] === |
|||
[[File:FreeBurnsideGroupExp3Gens2.png|thumb|350px|right|The [[free Burnside group]] <math>B(2,3)</math> is finite; in its [[Cayley graph]], shown here, each of its 27 elements is represented by a vertex. The question of which other groups <math>B(m,n)</math> are finite remains open.]] |
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* Is every [[finitely presented group|finitely presented]] [[periodic group]] finite? |
|||
* The [[inverse Galois problem]]: is every finite group the Galois group of a Galois extension of the rationals? |
|||
* For which positive integers ''m'', ''n'' is the [[free Burnside group]] {{nowrap|B(''m'',''n'')}} finite? In particular, is {{nowrap|B(2, 5)}} finite? |
|||
* Is every group [[surjunctive group|surjunctive]]? |
|||
* [[Andrews–Curtis conjecture]] |
|||
* [[Herzog–Schönheim conjecture]] |
|||
* Does [[Monstrous moonshine#Generalized moonshine|generalized moonshine]] exist? |
|||
* Are there an infinite number of [[Leinster group]]s? |
|||
* [[Guralnick–Thompson conjecture]]<ref>{{citation |last=Aschbacher |first=Michael |author-link=Michael Aschbacher |title=On Conjectures of Guralnick and Thompson |journal=[[Journal of Algebra]] |volume=135 |issue=2 |pages=277–343 |year=1990 |doi=10.1016/0021-8693(90)90292-V}}</ref> |
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* [[Problems in loop theory and quasigroup theory]] consider generalizations of groups |
|||
* The [[Kourovka Notebook]] is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.<ref>{{citation |
|||
| last1 = Khukhro | first1 = Evgeny I. |
|||
| last2 = Mazurov | first2 = Victor D. |author-link2 = Victor Mazurov |
|||
| arxiv = 1401.0300v16 |
|||
| title = Unsolved Problems in Group Theory. The Kourovka Notebook |
|||
| year = 2019}}</ref> |
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=== [[Teori model]] and [[bahasa formal]] === |
|||
* [[Vaught's conjecture]] |
|||
* The [[Stable group|Cherlin–Zilber conjecture]]: A simple group whose first-order theory is [[Stable theory|stable]] in <math>\aleph_0</math> is a simple algebraic group over an algebraically closed field. |
|||
* The Main Gap conjecture, e.g. for uncountable [[First order theory|first order theories]], for [[Abstract elementary class|AECs]], and for <math>\aleph_1</math>-saturated models of a countable theory.<ref name=":0">Shelah S, ''Classification Theory'', North-Holland, 1990</ref> |
|||
* Determine the structure of Keisler's order<ref>{{cite journal | last1 = Keisler | first1 = HJ | year = 1967 | title = Ultraproducts which are not saturated | journal = J. Symb. Log. | volume = 32 | issue = 1| pages = 23–46 | doi=10.2307/2271240| jstor = 2271240 }}</ref><ref>[[Maryanthe Malliaris|Malliaris M]], [[Saharon Shelah|Shelah S]], "A dividing line in simple unstable theories." https://arxiv.org/abs/1208.2140 {{Webarchive|url=https://web.archive.org/web/20170802171447/https://arxiv.org/abs/1208.2140 |date=2017-08-02 }}</ref> |
|||
* The stable field conjecture: every infinite field with a [[Stable theory|stable]] first-order theory is separably closed. |
|||
* Is the theory of the field of Laurent series over <math>\mathbb{Z}_p</math> [[Decidability (logic)|decidable]]? of the field of polynomials over <math>\mathbb{C}</math>? |
|||
* (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?<ref>Gurevich, Yuri, "Monadic Second-Order Theories," in [[Jon Barwise|J. Barwise]], [[Solomon Feferman|S. Feferman]], eds., ''Model-Theoretic Logics'' (New York: Springer-Verlag, 1985), 479–506.</ref> |
|||
* The Stable Forking Conjecture for simple theories<ref>{{cite journal | last1 = Peretz | first1 = Assaf | year = 2006 | title = Geometry of forking in simple theories | journal = Journal of Symbolic Logic| volume = 71 | issue = 1| pages = 347–359 | doi=10.2178/jsl/1140641179| arxiv = math/0412356| s2cid = 9380215 }}</ref> |
|||
* For which number fields does [[Hilbert's tenth problem]] hold? |
|||
* Assume K is the class of models of a countable first order theory omitting countably many [[Type (model theory)|types]]. If K has a model of cardinality <math>\aleph_{\omega_1}</math> does it have a model of cardinality continuum?<ref>{{cite journal |last=Shelah |first=Saharon |author-link=Saharon Shelah |date=1999 |title=Borel sets with large squares |journal=[[Fundamenta Mathematicae]] |arxiv=math/9802134 |volume=159 |issue=1 |pages=1–50|bibcode=1998math......2134S |doi=10.4064/fm-159-1-1-50 |s2cid=8846429 }}</ref> |
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* Shelah's eventual categoricity conjecture: For every cardinal <math>\lambda</math> there exists a cardinal <math>\mu(\lambda)</math> such that If an [[Abstract elementary class|AEC]] K with LS(K)<= <math>\lambda</math> is categorical in a cardinal above <math>\mu(\lambda)</math> then it is categorical in all cardinals above <math>\mu(\lambda)</math>.<ref name=":0" /><ref>{{Cite book |
|||
| title = Classification theory for abstract elementary classes |
|||
| last = Shelah |
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| first = Saharon |
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| publisher = College Publications |
|||
| year = 2009 |
|||
| isbn = 978-1-904987-71-0 |
|||
}}</ref> |
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* Shelah's categoricity conjecture for <math>L_{\omega_1,\omega}</math>: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.<ref name=":0" /> |
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* Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?<ref>Makowsky J, "Compactness, embeddings and definability," in ''Model-Theoretic Logics'', eds Barwise and Feferman, Springer 1985 pps. 645–715.</ref> |
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* If the class of atomic models of a complete first order theory is [[Categorical (model theory)|categorical]] in the <math>\aleph_n</math>, is it categorical in every cardinal?<ref>{{cite book |last=Baldwin |first=John T. |date=July 24, 2009 |title=Categoricity |publisher=[[American Mathematical Society]] |isbn=978-0-8218-4893-7 |url=http://www.math.uic.edu/~jbaldwin/pub/AEClec.pdf |access-date=February 20, 2014 |archive-url=https://web.archive.org/web/20100729073738/http://www.math.uic.edu/%7Ejbaldwin/pub/AEClec.pdf |archive-date=July 29, 2010 |url-status=live }}</ref><ref>{{cite journal |last=Shelah |first=Saharon |title=Introduction to classification theory for abstract elementary classes |url=http://front.math.ucdavis.edu/0903.3428|bibcode=2009arXiv0903.3428S |year=2009 |arxiv=0903.3428 }}</ref> |
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* Is every infinite, minimal field of characteristic zero [[algebraically closed field|algebraically closed]]? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.) |
|||
* Kueker's conjecture<ref>{{cite journal | last1 = Hrushovski | first1 = Ehud | year = 1989 | title = Kueker's conjecture for stable theories | journal = Journal of Symbolic Logic | volume = 54 | issue = 1| pages = 207–220 | doi=10.2307/2275025| jstor = 2275025 }}</ref> |
|||
* Does there exist an [[o-minimal]] first order theory with a trans-exponential (rapid growth) function? |
|||
* Does a finitely presented homogeneous structure for a finite relational language have finitely many [[reduct]]s? |
|||
* Do the [[Henson graph]]s have the [[finite model property]]? |
|||
* The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?<ref>{{cite journal |last1=Cherlin |first1=G. |last2=Shelah |first2=S. |date=May 2007 |title=Universal graphs with a forbidden subtree |journal=[[Journal of Combinatorial Theory, Series B]] |arxiv=math/0512218 |doi=10.1016/j.jctb.2006.05.008 |volume=97 |issue=3 |pages=293–333|s2cid=10425739 }}</ref> |
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* The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?<ref>Džamonja, Mirna, "Club guessing and the universal models." ''On PCF'', ed. M. Foreman, (Banff, Alberta, 2004).</ref> |
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* [[Generalized star height problem]] |
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* [[Tarski's exponential function problem]] |
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=== [[Teori bilangan]] === |
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==== Umum ==== |
|||
[[File:Perfect number Cuisenaire rods 6.png|thumb|6 is a [[perfect number]] because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them are odd.]] |
|||
*[[Grand Riemann hypothesis]] |
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**[[Generalized Riemann hypothesis]] |
|||
***[[Riemann hypothesis]] |
|||
* [[n conjecture|''n'' conjecture]] |
|||
** [[abc conjecture|''abc'' conjecture]] |
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** [[Szpiro's conjecture]] |
|||
* [[Hilbert's ninth problem]] |
|||
* [[Hilbert's eleventh problem]] |
|||
* [[Hilbert's twelfth problem]] |
|||
* [[Carmichael's totient function conjecture]] |
|||
* [[Erdős–Straus conjecture]] |
|||
* [[Erdős–Ulam problem]] |
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* [[Pillai's conjecture]] |
|||
* [[Hall's conjecture]] |
|||
* [[Lindelöf hypothesis]] and its consequence, the [[density hypothesis]] for zeroes of the Riemann zeta function (see [[Bombieri–Vinogradov theorem]]) |
|||
* [[Montgomery's pair correlation conjecture]] |
|||
* [[Hilbert–Pólya conjecture]] |
|||
* [[Grimm's conjecture]] |
|||
* [[Leopoldt's conjecture]] |
|||
* [[Scholz conjecture]] |
|||
* Do any [[odd perfect number]]s exist? |
|||
* Are there infinitely many [[perfect numbers]]? |
|||
* Do [[quasiperfect number]]s exist? |
|||
* Do any odd [[weird number]]s exist? |
|||
* Do any [[Lychrel number]]s exist? |
|||
* Is 10 a [[solitary number]]? |
|||
* [[aliquot sequence#Catalan-Dickson_conjecture|Catalan–Dickson conjecture on aliquot sequences]] |
|||
* Do any [[Generalized taxicab number|Taxicab(5, 2, n)]] exist for ''n'' > 1? |
|||
* [[Brocard's problem]]: existence of integers, (''n'',''m''), such that ''n''! + 1 = ''m''<sup>2</sup> other than ''n'' = 4, 5, 7 |
|||
* [[Beilinson conjecture]] |
|||
* [[Littlewood conjecture]] |
|||
* [[Vojta's conjecture]] |
|||
* [[Goormaghtigh conjecture]] |
|||
* [[Congruent number problem]] (a corollary to [[Birch and Swinnerton-Dyer conjecture]], per [[Tunnell's theorem]]) |
|||
* [[Lehmer's totient problem]]: if φ(''n'') divides ''n'' − 1, must ''n'' be prime? |
|||
* Are there infinitely many [[amicable numbers]]? |
|||
* Are there any pairs of [[amicable numbers]] which have opposite parity? |
|||
* Are there any pairs of [[relatively prime]] [[amicable numbers]]? |
|||
* Are there infinitely many [[betrothed numbers]]? |
|||
* Are there any pairs of [[betrothed numbers]] which have same parity? |
|||
* The [[Gauss circle problem]] – how far can the number of integer points in a circle centered at the origin be from the area of the circle? |
|||
* [[Divisor summatory function#Piltz divisor problem|Piltz divisor problem]], especially [[Dirichlet's divisor problem]] |
|||
* [[Van der Corput's method#Exponent pairs|Exponent pair conjecture]] |
|||
* Is π a [[normal number]] (its digits are "random")?<ref>{{cite web|url=http://www2.lbl.gov/Science-Articles/Archive/pi-random.html|title=Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key|access-date=2016-03-18|archive-url=https://web.archive.org/web/20160327035021/http://www2.lbl.gov/Science-Articles/Archive/pi-random.html|archive-date=2016-03-27|url-status=live}}</ref> |
|||
* [[Casas-Alvero conjecture]] |
|||
* [[Sato–Tate conjecture]] |
|||
* Find value of [[De Bruijn–Newman constant]] |
|||
* Which integers can be written as the [[Sums of three cubes|sum of three perfect cubes]]?<ref>{{Cite arxiv |eprint = 1604.07746v1|last1 = Bruhn|first1 = Henning|title = Newer sums of three cubes|last2 = Schaudt|first2 = Oliver|class = math.NT|year = 2016}}</ref> |
|||
* Erdős–Moser problem: is 1<sup>1</sup> + 2<sup>1</sup> = 3<sup>1</sup> the only solution to the [[Erdős–Moser equation]]? |
|||
* Is there a [[covering system]] with odd distinct moduli?<ref>{{citation |
|||
| last1 = Guo | first1 = Song |
|||
| last2 = Sun | first2 = Zhi-Wei |
|||
| doi = 10.1016/j.aam.2005.01.004 |
|||
| issue = 2 |
|||
| journal = Advances in Applied Mathematics |
|||
| mr = 2152886 |
|||
| pages = 182–187 |
|||
| title = On odd covering systems with distinct moduli |
|||
| volume = 35 |
|||
| year = 2005| arxiv = math/0412217 |
|||
| s2cid = 835158 |
|||
}}</ref> |
|||
* [[Singmaster's conjecture]]: is there a finite upper bound on the multiplicities of the entries greater than 1 in [[Pascal's triangle]]?<ref>{{citation |
|||
| last = Singmaster | first = D. | author-link = David Singmaster |
|||
| doi = 10.2307/2316907 |
|||
| mr = 1536288 |
|||
| issue = 4 |
|||
| journal = [[American Mathematical Monthly]] |
|||
| pages = 385–386 |
|||
| title = Research Problems: How often does an integer occur as a binomial coefficient? |
|||
| volume = 78 |
|||
| year = 1971 |
|||
| jstor = 2316907}}.</ref> |
|||
* The [[Markov number#Other properties|uniqueness conjecture for Markov numbers]]<ref>{{citation |
|||
| last = Aigner | first = Martin |
|||
| doi = 10.1007/978-3-319-00888-2 |
|||
| isbn = 978-3-319-00887-5 |
|||
| location = Cham |
|||
| mr = 3098784 |
|||
| publisher = Springer |
|||
| title = Markov's theorem and 100 years of the uniqueness conjecture |
|||
| year = 2013}}</ref> |
|||
* [[Keating–Snaith conjecture]] concerning the asymptotics of an integral involving the Riemann zeta function<ref>{{citation |
|||
|last=Conrey |first=Brian |author-link=Brian Conrey |
|||
|doi=10.1090/bull/1525 |
|||
|title=Lectures on the Riemann zeta function (book review) |
|||
|journal=[[Bulletin of the American Mathematical Society]] |
|||
|volume=53 |number=3 |pages=507–512 |year=2016|doi-access=free}}</ref> |
|||
* [[Newman's conjecture]] |
|||
==== [[Teori bilangan aditif]] ==== |
|||
{{See also|Masalah yang melibatkan barisan arimetik}} |
|||
* [[Konjektur Beal]] |
|||
* [[Konjektur Fermat–Catalan]] |
|||
* [[Konjektur Goldbach]] |
|||
* [[Konjektur Lemoine]] |
|||
* Nilai <math>g(k)</math> dan <math>G(k)</math> dalam [[masalah Waring]] |
|||
* [[Konjektur Lander, Parkin, dan Selfridge]] |
|||
* [[Konjektur Gilbreath]] |
|||
* [[Konjektur Erdős pada barisan aritmetik]] |
|||
* [[Konjektur Erdős–Turán pada dasar aditif]] |
|||
* [[Konjektur bilangan oktahedral Pollock]] |
|||
* [[Masalah Skolem]] |
|||
* Menentukan laju pertumbuhan <math>r_k(N)</math> (lihat [[teorema Szemerédi]]) |
|||
* [[Masalah bertindih minimum]] |
|||
* Apakah [[bilangan Ulam]] memiliki sebuah kerapatan positif? |
|||
==== [[Teori bilangan aljabar]] ==== |
|||
* Apakah terdapat banyaknya [[Masalah bilangan kelas#Medan kuadrat real|medan bilangan kuadrat]] dengan [[faktorisasi tunggal]] ([[Masalah bilangan kelas]])? |
|||
* Mencirikan semua medan bilangna aljabar yang memiliki suatu basis pangkatCharacterize all algebraic number fields that have some [[Algebraic number field#Bases for number fields|power basis]]. |
|||
* [[Konjektur Stark]] (termasuk [[konjektur Brumer–Stark]]) |
|||
* [[Konjektur Kummer–Vandiver]] |
|||
* [[Konjektur Greenberg]] |
|||
* [[Masalah Hermite]] |
|||
====[[Teori bilangan komputasi]]==== |
|||
* [[Faktorisasi bilangan bulat]]: Dapatkah faktorisasi bilangan bulat diselesaikan dalam waktu polinomial? |
|||
==== [[Bilangan prima]] ==== |
|||
{{Konjektur bilangan prima}} |
|||
[[Image:Goldbach partitions of the even integers from 4 to 50 rev4b.svg|thumb=Goldbach_partitions_of_the_even_integers_from_4_to_28_300px.png|300px|[[Goldbach's conjecture]] states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.]] |
|||
* [[Konjektur Goldbach]] |
|||
* [[Konjektur prima kembar]] |
|||
* [[Konjektur Polignac]] |
|||
* [[Konjektur Brocard]] |
|||
* [[Konjektur Mersenne Catalan]] |
|||
* [[Konjektur Agoh–Giuga]] |
|||
* [[Konjektur Dubner]] |
|||
* Masalah [[parit Gauss]]: apakah mungkin untuk menemukan sebuah barisan takhingga dari [[bilangan prima Gauss]] yang berbeda sehingga beda di antara bilangan berurutan dalam barisan adalah terbatas? |
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*[[Konjektur Mersenne#Konjektur Mersenne terbaru|Konjektur Mersenne terbaru]] |
|||
* [[Konjektur Erdős–Mollin–Walsh]] |
|||
* [[Konjektur Bunyakovsky]] |
|||
* [[Konjektur Dickson]] |
|||
* [[H hipotesis Schinzel]] |
|||
* Apakah terdapat [[kembar empat bilangan prima]] banyak? |
|||
* Apakah terdapat [[bilangan prima sepupu]] banyak? |
|||
* Apakah terdapat [[bilangan prima seksi]] banyak? |
|||
* Apakah terdapat [[bilangan prima Mersenne]] ([[konjektur Lenstra–Pomerance–Wagstaff]]); dengan setara, [[bilangan sempurna]] genap banyak? |
|||
* Apakah terdapat [[bilangan prima Wagstaff]] banyak? |
|||
* Apakah terdapat [[bilangan prima Sophie Germain]] banyak? |
|||
* Apakah terdapat [[bilangan prima Pierpont]]? |
|||
* Apakah terdapat [[bilangan prima beraturan]], dan jika begitu kerapatan nisbi <math>e^{-1/2}</math>? |
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* Untuk suatu bilangan bulat <math>b</math> yang bukan sebuah pangkat sempurna dan bukan dari bentuk <math>-4k^4</math> untuk bilangan bulat <math>k</math>, apakah terdapat bilangan prima [[satuan berulang]] banyak untuk basis <math>b</math>? |
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* Apakah terdapat [[bilangan prima Cullen]] banyak? |
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* Apakah terdapat [[bilangan prima Woodall]] banyak? |
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* Apakah terdapat [[bilangan prima Carol]] banyak? |
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* Apakah terdapat [[bilangan prima Kynea]] banyak? |
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* Apakah terdapat [[bilangna prima pandromik]] banyak setiap basis? |
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* Apakah terdapat [[bilangan prima Fibonacci]] banyak? |
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* Apakah terdapat [[bilangan prima Lucas]] banyak? |
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* Apakah terdapat [[bilangan prima Pell]] banyak? |
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* Apakah terdapat [[bilangan prima Newman–Shanks–Williams]] banyak? |
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* Apakah semua [[bilangan prima Mersenne]] dari [[Bilangan bulat kuadrat bebas|kuadrat bebas]] indeks? |
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* Apakah terdapat [[bilangan prima Weiferich]] banyak? |
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* Apakah terdapat suatu bilangan Wieferich dalam basis 47? |
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* Apakah terdapat suatu bilangan komposit <math>c</math> memenuhi <math>2^{c - 1} \equiv 1 \pmod {c^2}</math>? |
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* Untuk diberikan suatu bilangan bulat <math>a>0</math>, apakah terdapat bilangan prima <math>p</math> banyak sehingga <math>a^{p-1} \equiv 1 \pmod {p^2}</math>?<ref>{{cite book |last=Ribenboim |first=P. |author-link=Paulo Ribenboim |date=2006 |title=Die Welt der Primzahlen |edition=2nd |language=de |publisher=Springer |doi=10.1007/978-3-642-18079-8 |isbn=978-3-642-18078-1 |pages=242–243 |url=https://books.google.com/books?id=XMyzh-2SClUC&q=die+folgenden+probleme+sind+ungel%C3%B6st&pg=PA242|series=Springer-Lehrbuch }}</ref> |
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* Dapatkah sebuah bilangan prima <math>p</math> memenuhi <math>2^{p - 1} \equiv 1 \pmod {p^2}</math> dan <math>3^{p - 1} \equiv 1 \pmod {p^2}</math> secara serentak?<ref>{{cite arxiv |last=Dobson |first= J. B. |date=1 April 2017 |title=On Lerch's formula for the Fermat quotient |eprint=1103.3907v6|page=23|mode=cs2|class= math.NT }}</ref> |
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* Apakah terdapat [[bilangan prima Wilson]] banyak? |
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* Apakah terdapat [[bilangan prima Wolstenholme]] banyak? |
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* Apakah terdapat suatu [[bilangan prima Wall–Sun–Sun]]? |
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* Untuk diberikan suatu bilangan bulat <math>a>0</math>, apakah terdapat [[bilangan prima Lucas–Wieferich]] banyak terkait dengan pasangan <math>(a, -1)</math>? (Khususnya, ketika <math>a=1</math>, ini merupakan bilangan prima Fibonacci–Wieferich, dan ketika <math>a=2</math>, ini merupakan bilangan prima Pell–Wieferich) |
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* Apakah setiap [[bilangan Fermat]] <math>2^{2^n}+1</math> komposit untuk <math>n > 4</math>? |
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* Apakah semua bilangan Fermat [[Bilangan bulat kuadrat bebas|kuadrat bebas]]? |
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* Untuk suatu bilangan bulat <math>a</math> diberikan yang bukan sebuah kuadrat dan tidak sama dengan <math>-1</math>, apakah terdapat bilangan prima banyak dengan <math>a</math> sebagai sebuah akar primitif? |
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* [[Konjektur Artin pada akar primitf]] |
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* Apakah 78,557 merupakan [[bilangan Sierpiński]] terendah (disebut [[konjektur Selfridge]])? |
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* Apakah 509,203 merupakan [[bilangan Riesel]] terendah? |
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* Untuk sebuah bilangan bulat <math>k\ge 1</math>, <math>b \ge 2</math>, <math>c \ne 0</math> yang diberikan, dengan <math>\gcd(k,c) = 1</math> dan <math>\gcd(b,c) = 1</math>, apakah terdapat bilangan prima banyak dari bentuk <math display="inline">\frac{k \cdot b^n + c}{\gcd(k+c,b-1)}</math> dengan bilangan bulat <math>n \ge 1</math>? |
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* Konjektur Fortune (bahwa tidak ada [[bilangan Fortunate]] yang merupakan komposit) |
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* [[Masalah Landau]] |
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* [[Konjektur Feit–Thompson]] |
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* Apakah setiap bilangan prima muncul di [[barisan Euclid–Mullin]]? |
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* Apakah [[Teorema Wolstenholme#Sebalik sebagai sebuah dugaan|sebalik teorema Wolstenholme]] berlaku untuk semua bilangan asli? |
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* [[Konjektur Elliott–Halberstam]] |
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* Masalah yang terkait dengan [[teorema Linnik]] |
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* Mencari [[bilangan Skewes]] terkecil |
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=== [[Teori himpunan]] === |
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* Masalah mencari [[model teras]], salah satunya yang berisi semua [[Sifat kardinal besar|kardinal besar]]. |
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* Jika <math>\aleph_\omega</math> merupakan sebuah kardinal limit kuat, maka <math>2^{\aleph_\omega} < \aleph_{\omega_1}</math> (lihat [[Hipotesis kardinal tunggal]]). Batas terbaik, <math>\aleph_{\omega_4}</math>, diperoleh oleh [[saharon Shelah|Shelah]] menggunakan [[teori kofinalitas mungkin]]<nowiki/>nya. |
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* [[Logika-Ω|Hipotesis-Ω]] [[W. Hugh Woodin|Woodin]]. |
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* Apakah [[kekonsistenan]] dari keberadaan [[kardinal kompak kuat]] menyiratkan keberadaan konsisten dari sebuah [[kardinal superkompak]]? |
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* ([[W. Hugh Woodin|Woodin]]) Apakah [[Hipotesis Kontinum Rampat]] di bawah sebuah [[kardinal kompak kuat]] menyiratkan [[Hipotesis Kontinum Rampat]] di mana-mana? |
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* Apakah ada sebuah [[Kardinal Jónsson|aljabar Jónsson]] pada <math>\aleph_\omega</math>? |
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* Tanpa mengasumsi [[aksioma pemilihan]], dapatkah sebuah [[Kardinal Reinhardt|pembenaman elementer taktrivial]] <math>V \to V</math> ada? |
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* Apakah [[Hipotesis Kontinum Rampat]] memerlukan [[Diamondsuit|<math>{\diamondsuit(E^{\lambda^+}_{\operatorname{cf}(\lambda)}})</math>]] untuk setiap [[kardinal tunggal]] <math>\lambda</math>? |
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* Apakah [[Hipotesis Kontinum Rampat]] menyiratkan keberadaan [[Pohon Suslin|pohon Suslin-ℵ<sub>2</sub>]]? |
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*Aapakah [[aksioma pewarnaan buka]] konsisten dengan <math>2^{\aleph_{0}}>\aleph_{2}</math>? |
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===[[Topologi]]=== |
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[[Image:Ochiai_unknot.svg|right|thumb|250px|The [[unknotting problem]] asks whether there is an efficient algorithm to identify when the shape presented in a [[knot diagram]] is actually the [[unknot]].]] |
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*[[Konjektur Baum–Connes]] |
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*[[Konjektur Bing–Borsuk]] |
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* [[Konjektur Borel]] |
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* [[Konjektur Hilbert–Smith]] |
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* [[Konjektur Mazur]]<ref>{{citation|last=Mazur|first=Barry|author-link=Barry Mazur|title=The topology of rational points|journal=[[Experimental Mathematics (journal)|Experimental Mathematics]]|volume=1|number=1|year=1992|pages=35–45|doi=10.1080/10586458.1992.10504244|url=https://projecteuclid.org/euclid.em/1048709114|access-date=2019-04-07|archive-url=https://web.archive.org/web/20190407161124/https://projecteuclid.org/euclid.em/1048709114|archive-date=2019-04-07|url-status=live|doi-broken-date=2021-01-14}}</ref> |
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* [[Konjektur Novikov]] |
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*[[Konjektur Ravenel|Konjektur teropong]] |
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* [[Masalah takbuhulan]] |
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* [[Konjektur volume]] |
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* [[Konjektur Whitehead]] |
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* [[Konjektur Zeeman]] |
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=== Teori Grup === |
=== Teori Grup === |
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Baris 515: | Baris 929: | ||
=== Teori Himpunan === |
=== Teori Himpunan === |
||
=== |
=== Topolog === |
||
* Konjektur Baum-Connes |
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* Konjektur Borel |
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* Konjektur Hilbert-Smith |
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* Konjektur Mazur |
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* Konjektur Novikov |
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* Masalah ketidakterikatan(unknotting) |
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* Konjektur Volume |
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* Konjektur Whitehead |
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* Konjektur Zeeman |
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== Persoalan yang sudah dipecahkan sejak 1995 == |
== Persoalan yang sudah dipecahkan sejak 1995 == |
Revisi per 11 Maret 2021 10.58
Artikel atau sebagian dari artikel ini mungkin diterjemahkan dari List of unsolved problems in mathematics di en.wikipedia.org. Isinya masih belum akurat, karena bagian yang diterjemahkan masih perlu diperhalus dan disempurnakan. Jika Anda menguasai bahasa aslinya, harap pertimbangkan untuk menelusuri referensinya dan menyempurnakan terjemahan ini. Anda juga dapat ikut bergotong royong pada ProyekWiki Perbaikan Terjemahan. (Pesan ini dapat dihapus jika terjemahan dirasa sudah cukup tepat. Lihat pula: panduan penerjemahan artikel) |
Sejak zaman Renaisans, banyak persoalan matematika dari abad sebelumnya yang dipecahkan abad setelahnya, tetapi sampai sekarang masih banyak persoalan matematika, besar maupun kecil, bermunculan dan belum terpecahkan.[1] Persoalan-persoalan ini seringkali datang dari berbagai bidang, termasuk fisika, ilmu komputer, aljabar, analisis, kombinatorika, geometri aljabar, diferensial, diskret, dan Euklides, teori graf, grup, model, bilangan, himpunan dan Ramsey, sistem dinamikal, persamaan diferensial parsial, dan masih banyak lagi. Beberapa masalah memiliki lebih dari satu mata pelajaran matematika dan dipelajari menggunakan teknik-tennik dari bidang yang berbeda. Hadiahnya seringkali dberikan untuk penyelesaian ke sebuah masalah yang lama, dan daftar-daftatr persoaln yang belum terpecahkan (seperti daftar Masalah Hadiah Millenium) menerima banyak perhatian.
Artikel ini merupakan sebuah gabungan masalah yang belum terpecahkan yang diturunkan dari banyak sumber, termasuk namun tidak terbatas pada daftar-daftar dianggap berwibawa, ini mungkin tidak selalu mutakhir, dan ini termasuk masalah yang dianggap oleh komunitas matematika menjadi sangat bervariasi dalam kesulitan dan sentralitas ilmu pengetahuan secara keseluruhan.
Artikel ini mengumpulkan berbagai persoalan yang didapat dari berbagai sumber. Daftar ini belum tentu lengkap atau terbarukan
Daftar masalah yang belum terpecahkan dalam matematika
Berbagai matematikawan dan organisasi telah menerbitkan dan mendukung daftar persoalan matematika yang belum terpecahkan. Dalam beberapa kasus, daftar tersebut telah berkaitan dengan hadiah-hadiah untuk penemuan-penemuan penyelesaiannya.
Daftar | Jumlah masalah | Jumlah yang belum terpecahkan atau belum terselesaikan sepenuhnya | Diusulkan oleh | Diusulkan pada tahun |
---|---|---|---|---|
Masalah Hilbert[2] | 23 | 15 | David Hilbert | 1900 |
Masalah Landau[3] | 4 | 4 | Edmund Landau | 1912 |
Masalah Tanimaya[4] | 36 | - | Yutaka Taniyama | 1955 |
24 pertanyaan Thurston[5][6] | 24 | - | William Thurston | 1982 |
Masalah Smale | 18 | 14 | Stephen Smale | 1998 |
Masalah Hadiah Millenium | 7 | 6[7] | Clay Mathematics Institute | 2000 |
Masalah Simon | 15 | <12[8][9] | Barry Simon | 2000 |
Masalah yang Belum Terpecahkan pada Matematika untuk Abad ke-21[10] | 22 | - | Jair Minoro Abe, Shotaro Tanaka | 2001 |
Tantangan matematika DARPA[11][12] | 23 | - | DARPA | 2007 |
Masalah Hadiah Millenium
Dari tujuh Masalah Hadiah Millenium asli diatur oleh Clay Mathematics Institute pada tahun 2000, keenam masalah telah belum dipecahkan pada Juli, 2020.[13]
- Masalah P versus NP
- Konjektur Hodge
- Hipotesis Riemann
- Keberadaan Yang–Mills dan sela massa
- Keberadaan Navier–Stokes dan kemulusan
- Konjektur Birch dan Swinnerton-Dyer
Masalah ketujuh, konjektur Poincaré, telah dipecahkan,[14] namun, sebuah rampat disebut konjektur Poincaré empat dimensi mulus—yaitu, apakah sebuah bola topologi empat dimensi dapat memiliki dua struktur mulus yang tidak setara atau lebih—masih belum terpecahkan.[15]
Masalah yang belum terpecahkan
Aljabar
- Konjektur homologis dalam aljabar komutatif
- Masalah wakilan kekisi hingga
- Masalah keenambelas Hilbert
- Masalah kelimabelas Hilbert
- Konjektur Hadamard
- Konjektur Jacobson
- Konjektur Crouzeix
- Keberadaan kuboid sempurna dan konjektur kuboid yang terkait
- Konjektur Zauner: keberadaan SIC-POVM di semua dimensi
- Masalah liar: Penggolongan pasangan matriks dalam konjugasi simultan dan masalah yang mengandungnya seperti banyak masalah penggolongan
- Konjektur Köthe
- Konjektur Birch–Tate
- Konjektur II Serre
- Konjektur Bombien–Lang
- Konjektur Farrell–Jones
- Konjektur Bost
- Konjektur basis Rota
- Konjektur seragam
- Konjektur Kaplansky
- Konjektur Kummer–Vandiver
- Konjektur kegandaan Serre
- Konjektur Pierce–Birkhoff
- Konjektur Eilenberg–Ganea
- Konjektur Green
- Konjektur kelengkungan-p Grothendieck–Katz
- Konjektur Sendov
- Konjektur Zariski–Lipman
- Buku Catatan Dneister (Dnestrovskaya Tetrad) mengumpulkan beberapa ratusan masalah-masalah yang belum terpecahkan dalam aljabar, khususnya teori gelanggang dan teori modulus.[16]
- Buku Catatan Erlagol (Erlagolskaya Tetrad) mengumpulkan masalah-masalah yang belum terpecahkan dalam aljabar dan teori model.[17]
Analisis
- Empat konjektur eksponensial pada transenden setidaknya salah satu dari empat eksponensial gabungan irasional[18]
- Konjektur Lehmer pada ukuran polinomial siklotomik Mahler[19]
- Masalah Pompeiu pada topologi domain untuk yang beberapa fungsi taknol memiliki integral lenyap pada setiap salinan kongruen[20]
- Konjektur Schanuel pada derajat transenden dari eksponensial irasoinal bebas linear[21]
- Apakah (konstanta Euler–Mascheroni), , , , , , , , , , , , konstanta Catalan, atau konstanta Khinchin rasional, irasional aljabar, atau transendental? Berapa ukuran keirasionalan dari setiap bilangan-bilangan ini?[22][23][24]
- Konjektur Vitushkin
- Masalah subruang invarian
- Konjektur Kung–Traub[25]
- Keteraturan dari penyelesaian persamaan Vlasov–Maxwell
- Keteraturan dari penyelesaian persamaan Euler
- Kekonvergenan deret Flint Hills
Kombinatorika
- Konjektur himpunan gabungan tertutup Franki: untuk setiap keluarga himpunan ditutup dalam jumlah, terdapat sebuah elemen (dari ruang pendasar) milik setengah atau lebih dari himpunan-himpunan tersebut[26]
- Konjektur pelari kesepian: jika pelari berpasangan dengan kecepatan yang berbeda berlari mengitari lintasan panjang satuan, apakah setiap pelari akan "kesepian" (yaitu, setidaknya sebuah jarak dari setiap pelari lainnya) pada suatu waktu?[27]
- Mencari sebuah fungsi untuk memodelkan n-langkah langkah hindar-diri[28]
- Konjektur 1/3–2/3: apakah setiap himpunan terurut parsial terhingga yang bukan terurut total berisi dua elemen dan sehingga probabilitasnya bahwa sebelum dalam sebuah pengembangan linear acak di antara 1/3 dan 2/3?[29]
- Mmeberikan sebuah interpretasi kombinatorial dari koefisien Kronecker.[30]
- Pertanyaan terbuka mengenai persegi Latin
- Nilai dari bilangan Dedekind untuk .[31]
- Nilai dari bilangan Ramsey, khususnya
- Nilai dari bilangan Van der Waerden
Sistem dinamikal
- Konjektur Collatz (konjektur )
- Metode kedua Lyapunov untuk kestabilan – Untuk apa kelas persamaan diferensial biasa, yang menjelaskan sistem dinamika, apakah metode kedua Lyapunov yang dirumuskan dalam bentuk klasik dan kekanonisan yang dirampat menentukan syarat perlu dan cukup untuk kestabilan (asimtotis) gerak?
- Konjektur Furstenberg – apakah setipa ukura nyang invarian dan ergodik untuk tindakan , pada lingkaran Lebesgue atau atomik?
- Konjektur Margulis – Pengglongan ukuran untuk tindakan terdiagonalkan dalam grup peringkat tinggi
- Konjektur MLC – apakah himpunan Mandelbrot terhubung lokal?
- Konjektur Weinstein – Apakah sebuah himpunan aras tipe kontak kompak beraturan dari sebuah Hamilton pada sebuah manifold simplektik membawa setidaknya satu orbit berkala dari alir Hamilton?
- Konjektur Arnold–Givental dan konjektur Arnold – berkaitan geometri simplektik dengan teori Morse
- Konjektur Eremenko bahwa setiap komponen dari himpunan pelepasan sebuah fungsi transendental menyeluruh tidak terbatas
- Apakah setiap automaton seluler terbalikkan dalam tiga dimensi atau lebih secara lokal terbalikkan?[32]
- Konjektur Birkhoff: jika sebuah tabel terintegralkan dan cembung sempurna, apakah batasnya yang semestinya sebuah elips?[33]
- Banyak masalah berkaitan dengan sebuah biliar luar, sebagai contoh menunjukkan bahwa biliar luar relatif dengan hampir setiap poligon cembung memiliki orbit-orbit yang tidak terbatas.
- Konjektur ergodisitas tunggal kuantum[34]
- Konjektur Berry–Tabor
- Konjektur Painlevé
Permainan dan teka-teki
Permainan kombinatorial
- Sudoku:
- Berapa jumlah maksimum yang diberikan untuk sebuah teka-teki minimal?[35]
- Berapa banyak teka-teki yang seharusnya memiliki satu penyelesaian?[36]
- Berapa banyak teka-teki dengan tepatnya satu penyelesaian merupakan minimal[37]
- Variasi silang-bulat-silang:
- Diberikan sebuah lebar papan silang-bulat-silang, berapa dimensi paling terkecil sehingga dijamin sebuah strategi kemenangan?[38]
- Apa status kelengkapan Turing dari semua Permainan dengan tunggal?
Permainan dengan informasi yang tidak sempurna
Geometri
Geometri aljabar
- Konjektur limpahan
- Konjektur Bass
- Konjektur Deligne
- Konjektur Dixmier
- Konjektur Fröberg
- Konjektur Fujita
- Konjektur Hartshorne[39]
- Konjektur Jacobi
- Konjektur Manin
- Konjektur Maulik–Nekrasov–Okounkov–Pandharipande pada sebuah kesetaraan antara teorema Gromov–Witten and teorema Donaldson–Thomas [40]
- Konjektur Nakai
- Resolusi kesingularan dalam karateristik
- Konjektur standar pada siklus aljabar
- Konjektur bagian
- Konjektur Tate
- Penghentian pembalikan
- Konjektur Virasoro
- Konjektur monodromi bobot
- Konjektur kegandaan Zariski[41]
Peliputan dan pengepakan
- Masalah Borsuk pada batas atas dan bawah untuk bilangan himpunan bagian dimater yang terkecil dibutuhkan menjadi sebuah himpunan dimensi terbatas.
- Masalah pengepakan Rado: jika gabungan persegi yang bayak memilki luas satuan, seberpa kecil dapat luas terbesaar diliputi oelh sebuah himpunan bagian lepas persegi-persegi?[42]
- Konjektur Erdős–Oler yang ketika merupakan sebuah bilangan segitiga, pengepakan lingkaran dalam sebuah segitiga sama sisi membutuhkan sebuah segitiga dari ukuran yang sama sebagai pengepakan lingkaran [43]
- Masalah bilangan ciuman untuk dimensi selain 1, 2, 3, 4, 8 dan 24[44]
- Konjektur Reinhardt bahwa oktagon yang mulus memiliki keraptan pengepakan maksimum terendah dari semua himpunan bidang simetris pusat[45]
- Masalahpengepakan bola, termasuk kerapatan dari pengepakan terapat dalam dimensi selain 1, 2, 3, 8, dan 24, dan perilaku asimtotiknya untuk dimensi yang tinggi.
- Pengepakan persegi dalam sebuah persegi: berapa rata-rata pertumbuhan asimtotik dari ruang yang terbuang?[46]
- Konjektur pengepakan Ulam mengenai identitas dari padatan cembung pengepakan terburuk[47]
Geometri diferensial
- The Konjektur luas pengisi, yang sebuah setengah bola memiliki luas minimum disekitar among permukaan bebas pintas dalam ruang Euklides yang perbatasannya membentuk sebuah kurva tertutub dari panjang yang diberikan[48]
- Konjektur Hopf mengaitkan kelengkungan dan karaterisitk Euler dari manifold Riemann dimensi yang lebih tinggi[49]
- The Masalah Bernstein bola, sebuah rampat kemungkinan dari [Maslah Bernstein]] yang asli.
- Konjektur Cartan–Hadamard: Dapatkah pertidaksamaan isoperimetrik klasik untuk himpunan bagian ruang Euklides diperpanjang menjadi ruang kelengkungan takpositif, dikenal sebagai manifold Cartan–Hadamard?
- Konjektur Carathéodory
- Konjektur Chern (geometri afin)
- Konjektur Chern untuk hiperpermukaan dalam bola
- Konjektur Yau
- Konjektu Yau pada eigenniiai pertama
- Masalah kurva tertutup: Carilah syarat perlu dan cukup (eksplisit) yang menentukan ketika, diberikan dua fungsi berkalai dengan periode yang sama, kurva integral tertutup.[50]
Geometri diskret
- Menyelesaikan masalah akhir yang bahagia untuk sembarang [51]
- Mencari pemadanan batas atas dan bawah untuk himpunan-k dan membagi garis[52]
- Konjektur Hadwiger pada peliputan benda cembung n-dimensi dengan paling banyak salinan yang lebih kecil?[53]
- Masalah segitiga Kobon pada segitiga dalam garis urutan garis[54]
- Masalah Kusner yang paling banyak titik dapat berjarak sama dalam ruang [55]
- Masalah McMullen pada himpunan transformasi dengan cara proyeksi dari dua titik menjadi posisi cekung[56]
- Pengepakan penyanggah berkaki tiga[57]
- Berapa banyak jarak satuan yang dapat ditentukan oleh sebuah himpunan dari titik dalam bidang Euclides?[58]
- Masalah hutan buram
- Meningkatkan batas bawah dan atas untuk masalah segitiga Heilbronn.
- Konjektur 3^d Kalai pada jumlah kemungkinan terkecil dari sisi politop simetrik terpusat.[59]
Geometri Euklides
- Konjektur Atiyah pada konfigurasi[60]
- Belmann tersesat dalam sebuah hutan – carilah jalan terpendek yang dijamin mendekati batasnya dari sebuah bentuk yang diberikan, dimulai pada titik yang takdiketahui dari bentuk dengna orientasi yang takdiketahui[61]
- Gelanggang Borromean — apakah tiga kurva ruang taktersimpul, bukan semua tiga lingkaran, yang tidak dapat disusun untuk membentuk tautan ini?[62]
- Masalah Danzer dan masalah lalat mati Conway – apakah himpunan Danzer dari kerapatan yang dibatasi atau pemisahan yang dibatasi ada?[63]
- Pembedahan ke ortoskema – apakah mungkin untuk is it possible untuk simpleks-simpleks dari setiap dimensi?[64]
- Masalah einstein – apakah terdapat sebuah bentuk dua dimensi yang membentuk prototile untuk sebuah pengubinan aperiodik, tapi bukan untuk suatu pengubinan periodik?[65]
- Konjektur Falconer bahwa himpunan dimensi Hausdorff lebih besar daripada di harus memiliki sebuah himpunan jarak ukuran Lebesgue[66]
- Masalah persegi dalam, juga dikenal sebagai konjektur Toeplitz – apakah setiap kurva Jordan memilik sebuah persegi dalam?[67]
- Konjektur Kakeya – apakah himpunan -dimensi yang berisi sebuah ruas garis satuan dalam setiap arah selalu memiliki dimensi Hausdorff dan dimensi Minkowski sama dengan ?[68]
- Masalah Kelvin pada partisi luas permukaan minimum dari ruang ke sel volume yang sama, dan and the optimalitas dari struktur Weaire–Phelan sebagai sebuah penyelesaian untuk masalah Kelvin[69]
- Masalah peliputan semesta Lebesgue pada bentuk cembung luas minimum dalam bidang yang dapat meliputi suatu bentuk diameter[70]
- Konjektur Mahler pada darab dari volume benda cembung simetrik terpusat dan polarnya.[71]
- Masalah cacing Moser – berapakah luasterkecil dari sebuah bentuk yang dapat meliputi setiap kurva panjang satuan dalam bidang?[72]
- Masalah sofa bergerak – berapa luas terbesar dari sebuah bentuk yang dapat diarahkan melalui sebuah lebar satuan koridor berbentuk huruf L?[73]
- Masalah Shephard (atau konjektur Dürer) – apakah setiap polihedron cembung memiliki sebuah jaring, atau pembukaan lipatan tepi yang sederhana?[74][75]
- Masalah Thomson – berapa konfigurasi energi minimum dari partikel pengelakan satu sama lain pada sebuah bola satuan?[76]
- Seragam 5 politop – carilah dan golongkan himpunan sempurna dari bentuk-bentuk ini[77]
Teori graf
Lintasan dan siklus dalam graf
- Konjektur Barnette bahwa setiap graf planar tiga terhubung dwipihak kubik memiliki sebuah siklus Hamilton[78]
- Konjektur kekerasan Chvátal, bahwa terdapat sebuah bilangan sehingga setiap graf keras- adalah Hamilton[79]
- Konjektur peliputan ganda siklus bahwa setiap yang tak memiliki jembatan, memiliki sebuah keluarga siklus yang termasuk setiap tepi dua kali[80]
- Konjektur Erdős–Gyárfás pada siklus dengan panjang pangkat dari dua dalam graf kubik[81]
- Konjektur arborisitas linear pada penguraian graf menjadi gabungan lepas lintasan menurut derajat maksimumnya[82]
- Konjektur Lovász pada lintasan Hamilton dalam graf simetrik[83]
- Masalah Oberwolfach di mana 2 graf beraturan memilik sifat bahwa sebuah graf sempurna pada jumlah puncak yang sama dapat diuraikan menjadi salinan tepi-lepas dari graf yang diberikan.[84]
- Konjektur Szymanski
Pewarnaan and pelabelan graf
- Konjektur Cereceda pada diameter dari ruang pewarnaan graf merosot[85]
- Konjektur Erdős–Faber–Lovász pada gabungan pewarnaan klik[86]
- Konjektur Gyárfás–Sumner pada keterbatasan dari graf dengan sebuah pohon terimbas yang dliarang[87]
- Konjektur Hadwiger mengaitkan pewarnaan untuk minor klik[88]
- Masalah Hadwiger–Nelson pada bilangan kromatik dari graf jarak satuan[89]
- Konjektur pewarnaan Jaeger's Petersen bahwa setiap grafik kubik takberjembantan memiliki sebuah pemetaan siklus-kontinu ke graf Petersen[90]
- Daftar pewarnaan konjektur bahwa, untuk setiap grad, daftar kromatik indeks sama dengan indeks kromatik[91]
- Konjektur pewarnaan total Behzad dan Vizing bahwa bilangan kromatik total paling banyak dua ditambah derajat maksimum[92]
Graph drawing
- The Albertson conjecture that the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number[93]
- The Blankenship–Oporowski conjecture on the book thickness of subdivisions[94]
- Conway's thrackle conjecture[95]
- Harborth's conjecture that every planar graph can be drawn with integer edge lengths[96]
- Negami's conjecture on projective-plane embeddings of graphs with planar covers[97]
- The strong Papadimitriou–Ratajczak conjecture that every polyhedral graph has a convex greedy embedding[98]
- Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?[99]
- Universal point sets of subquadratic size for planar graphs[100]
Word-representation of graphs
- Characterise (non-)word-representable planar graphs [101][102][103][104]
- Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs [105])
- Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter [106]
- Is the line graph of a non-word-representable graph always non-word-representable? [101][102][103][104]
- Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter? [101][102][103][104]
- Is it true that out of all bipartite graphs crown graphs require longest word-representants? [107]
- Characterise word-representable graphs in terms of (induced) forbidden subgraphs. [101][102][103][104]
- Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)? [101][102][103][104]
Miscellaneous graph theory
- Conway's 99-graph problem: does there exist a strongly regular graph with parameters (99,14,1,2)?[108]
- The Erdős–Hajnal conjecture on large cliques or independent sets in graphs with a forbidden induced subgraph[109]
- The GNRS conjecture on whether minor-closed graph families have embeddings with bounded distortion[110]
- Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs[111]
- The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs[112]
- Jørgensen's conjecture that every 6-vertex-connected K6-minor-free graph is an apex graph[113]
- Meyniel's conjecture that cop number is [114]
- Does a Moore graph with girth 5 and degree 57 exist?[115]
- What is the largest possible pathwidth of an n-vertex cubic graph?[116]
- The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.[117][118]
- The second neighborhood problem: does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?[119]
- Do there exist infinitely many strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?[120]
- Sumner's conjecture: does every -vertex tournament contain as a subgraph every -vertex oriented tree?[121]
- Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every Petersen-minor-free bridgeless graph has a nowhere-zero 4-flow[122]
- Vizing's conjecture on the domination number of cartesian products of graphs[123]
- Zarankiewicz problem
Teori grup
- Is every finitely presented periodic group finite?
- The inverse Galois problem: is every finite group the Galois group of a Galois extension of the rationals?
- For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
- Is every group surjunctive?
- Andrews–Curtis conjecture
- Herzog–Schönheim conjecture
- Does generalized moonshine exist?
- Are there an infinite number of Leinster groups?
- Guralnick–Thompson conjecture[124]
- Problems in loop theory and quasigroup theory consider generalizations of groups
- The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.[125]
Teori model and bahasa formal
- Vaught's conjecture
- The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in is a simple algebraic group over an algebraically closed field.
- The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for -saturated models of a countable theory.[126]
- Determine the structure of Keisler's order[127][128]
- The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
- Is the theory of the field of Laurent series over decidable? of the field of polynomials over ?
- (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?[129]
- The Stable Forking Conjecture for simple theories[130]
- For which number fields does Hilbert's tenth problem hold?
- Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality does it have a model of cardinality continuum?[131]
- Shelah's eventual categoricity conjecture: For every cardinal there exists a cardinal such that If an AEC K with LS(K)<= is categorical in a cardinal above then it is categorical in all cardinals above .[126][132]
- Shelah's categoricity conjecture for : If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[126]
- Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[133]
- If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?[134][135]
- Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
- Kueker's conjecture[136]
- Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
- Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
- Do the Henson graphs have the finite model property?
- The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?[137]
- The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[138]
- Generalized star height problem
- Tarski's exponential function problem
Teori bilangan
Umum
- Grand Riemann hypothesis
- n conjecture
- Hilbert's ninth problem
- Hilbert's eleventh problem
- Hilbert's twelfth problem
- Carmichael's totient function conjecture
- Erdős–Straus conjecture
- Erdős–Ulam problem
- Pillai's conjecture
- Hall's conjecture
- Lindelöf hypothesis and its consequence, the density hypothesis for zeroes of the Riemann zeta function (see Bombieri–Vinogradov theorem)
- Montgomery's pair correlation conjecture
- Hilbert–Pólya conjecture
- Grimm's conjecture
- Leopoldt's conjecture
- Scholz conjecture
- Do any odd perfect numbers exist?
- Are there infinitely many perfect numbers?
- Do quasiperfect numbers exist?
- Do any odd weird numbers exist?
- Do any Lychrel numbers exist?
- Is 10 a solitary number?
- Catalan–Dickson conjecture on aliquot sequences
- Do any Taxicab(5, 2, n) exist for n > 1?
- Brocard's problem: existence of integers, (n,m), such that n! + 1 = m2 other than n = 4, 5, 7
- Beilinson conjecture
- Littlewood conjecture
- Vojta's conjecture
- Goormaghtigh conjecture
- Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem)
- Lehmer's totient problem: if φ(n) divides n − 1, must n be prime?
- Are there infinitely many amicable numbers?
- Are there any pairs of amicable numbers which have opposite parity?
- Are there any pairs of relatively prime amicable numbers?
- Are there infinitely many betrothed numbers?
- Are there any pairs of betrothed numbers which have same parity?
- The Gauss circle problem – how far can the number of integer points in a circle centered at the origin be from the area of the circle?
- Piltz divisor problem, especially Dirichlet's divisor problem
- Exponent pair conjecture
- Is π a normal number (its digits are "random")?[139]
- Casas-Alvero conjecture
- Sato–Tate conjecture
- Find value of De Bruijn–Newman constant
- Which integers can be written as the sum of three perfect cubes?[140]
- Erdős–Moser problem: is 11 + 21 = 31 the only solution to the Erdős–Moser equation?
- Is there a covering system with odd distinct moduli?[141]
- Singmaster's conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?[142]
- The uniqueness conjecture for Markov numbers[143]
- Keating–Snaith conjecture concerning the asymptotics of an integral involving the Riemann zeta function[144]
- Newman's conjecture
Teori bilangan aditif
- Konjektur Beal
- Konjektur Fermat–Catalan
- Konjektur Goldbach
- Konjektur Lemoine
- Nilai dan dalam masalah Waring
- Konjektur Lander, Parkin, dan Selfridge
- Konjektur Gilbreath
- Konjektur Erdős pada barisan aritmetik
- Konjektur Erdős–Turán pada dasar aditif
- Konjektur bilangan oktahedral Pollock
- Masalah Skolem
- Menentukan laju pertumbuhan (lihat teorema Szemerédi)
- Masalah bertindih minimum
- Apakah bilangan Ulam memiliki sebuah kerapatan positif?
Teori bilangan aljabar
- Apakah terdapat banyaknya medan bilangan kuadrat dengan faktorisasi tunggal (Masalah bilangan kelas)?
- Mencirikan semua medan bilangna aljabar yang memiliki suatu basis pangkatCharacterize all algebraic number fields that have some power basis.
- Konjektur Stark (termasuk konjektur Brumer–Stark)
- Konjektur Kummer–Vandiver
- Konjektur Greenberg
- Masalah Hermite
Teori bilangan komputasi
- Faktorisasi bilangan bulat: Dapatkah faktorisasi bilangan bulat diselesaikan dalam waktu polinomial?
Bilangan prima
- Konjektur Goldbach
- Konjektur prima kembar
- Konjektur Polignac
- Konjektur Brocard
- Konjektur Mersenne Catalan
- Konjektur Agoh–Giuga
- Konjektur Dubner
- Masalah parit Gauss: apakah mungkin untuk menemukan sebuah barisan takhingga dari bilangan prima Gauss yang berbeda sehingga beda di antara bilangan berurutan dalam barisan adalah terbatas?
- Konjektur Mersenne terbaru
- Konjektur Erdős–Mollin–Walsh
- Konjektur Bunyakovsky
- Konjektur Dickson
- H hipotesis Schinzel
- Apakah terdapat kembar empat bilangan prima banyak?
- Apakah terdapat bilangan prima sepupu banyak?
- Apakah terdapat bilangan prima seksi banyak?
- Apakah terdapat bilangan prima Mersenne (konjektur Lenstra–Pomerance–Wagstaff); dengan setara, bilangan sempurna genap banyak?
- Apakah terdapat bilangan prima Wagstaff banyak?
- Apakah terdapat bilangan prima Sophie Germain banyak?
- Apakah terdapat bilangan prima Pierpont?
- Apakah terdapat bilangan prima beraturan, dan jika begitu kerapatan nisbi ?
- Untuk suatu bilangan bulat yang bukan sebuah pangkat sempurna dan bukan dari bentuk untuk bilangan bulat , apakah terdapat bilangan prima satuan berulang banyak untuk basis ?
- Apakah terdapat bilangan prima Cullen banyak?
- Apakah terdapat bilangan prima Woodall banyak?
- Apakah terdapat bilangan prima Carol banyak?
- Apakah terdapat bilangan prima Kynea banyak?
- Apakah terdapat bilangna prima pandromik banyak setiap basis?
- Apakah terdapat bilangan prima Fibonacci banyak?
- Apakah terdapat bilangan prima Lucas banyak?
- Apakah terdapat bilangan prima Pell banyak?
- Apakah terdapat bilangan prima Newman–Shanks–Williams banyak?
- Apakah semua bilangan prima Mersenne dari kuadrat bebas indeks?
- Apakah terdapat bilangan prima Weiferich banyak?
- Apakah terdapat suatu bilangan Wieferich dalam basis 47?
- Apakah terdapat suatu bilangan komposit memenuhi ?
- Untuk diberikan suatu bilangan bulat , apakah terdapat bilangan prima banyak sehingga ?[145]
- Dapatkah sebuah bilangan prima memenuhi dan secara serentak?[146]
- Apakah terdapat bilangan prima Wilson banyak?
- Apakah terdapat bilangan prima Wolstenholme banyak?
- Apakah terdapat suatu bilangan prima Wall–Sun–Sun?
- Untuk diberikan suatu bilangan bulat , apakah terdapat bilangan prima Lucas–Wieferich banyak terkait dengan pasangan ? (Khususnya, ketika , ini merupakan bilangan prima Fibonacci–Wieferich, dan ketika , ini merupakan bilangan prima Pell–Wieferich)
- Apakah setiap bilangan Fermat komposit untuk ?
- Apakah semua bilangan Fermat kuadrat bebas?
- Untuk suatu bilangan bulat diberikan yang bukan sebuah kuadrat dan tidak sama dengan , apakah terdapat bilangan prima banyak dengan sebagai sebuah akar primitif?
- Konjektur Artin pada akar primitf
- Apakah 78,557 merupakan bilangan Sierpiński terendah (disebut konjektur Selfridge)?
- Apakah 509,203 merupakan bilangan Riesel terendah?
- Untuk sebuah bilangan bulat , , yang diberikan, dengan dan , apakah terdapat bilangan prima banyak dari bentuk dengan bilangan bulat ?
- Konjektur Fortune (bahwa tidak ada bilangan Fortunate yang merupakan komposit)
- Masalah Landau
- Konjektur Feit–Thompson
- Apakah setiap bilangan prima muncul di barisan Euclid–Mullin?
- Apakah sebalik teorema Wolstenholme berlaku untuk semua bilangan asli?
- Konjektur Elliott–Halberstam
- Masalah yang terkait dengan teorema Linnik
- Mencari bilangan Skewes terkecil
Teori himpunan
- Masalah mencari model teras, salah satunya yang berisi semua kardinal besar.
- Jika merupakan sebuah kardinal limit kuat, maka (lihat Hipotesis kardinal tunggal). Batas terbaik, , diperoleh oleh Shelah menggunakan teori kofinalitas mungkinnya.
- Hipotesis-Ω Woodin.
- Apakah kekonsistenan dari keberadaan kardinal kompak kuat menyiratkan keberadaan konsisten dari sebuah kardinal superkompak?
- (Woodin) Apakah Hipotesis Kontinum Rampat di bawah sebuah kardinal kompak kuat menyiratkan Hipotesis Kontinum Rampat di mana-mana?
- Apakah ada sebuah aljabar Jónsson pada ?
- Tanpa mengasumsi aksioma pemilihan, dapatkah sebuah pembenaman elementer taktrivial ada?
- Apakah Hipotesis Kontinum Rampat memerlukan untuk setiap kardinal tunggal ?
- Apakah Hipotesis Kontinum Rampat menyiratkan keberadaan pohon Suslin-ℵ2?
- Aapakah aksioma pewarnaan buka konsisten dengan ?
Topologi
- Konjektur Baum–Connes
- Konjektur Bing–Borsuk
- Konjektur Borel
- Konjektur Hilbert–Smith
- Konjektur Mazur[147]
- Konjektur Novikov
- Konjektur teropong
- Masalah takbuhulan
- Konjektur volume
- Konjektur Whitehead
- Konjektur Zeeman
Teori Grup
Teori Model dan bahasa formal
Teori Nomor
Teori Himpunan
Topolog
Persoalan yang sudah dipecahkan sejak 1995
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Lihat pula
Referensi
- ^ Eves, An Introduction to the History of Mathematics 6th Edition, Thomson, 1990, ISBN 978-0-03-029558-4.
- ^ Thiele, Rüdiger (2005), "On Hilbert and his twenty-four problems", dalam Van Brummelen, Glen, Mathematics and the historian's craft. The Kenneth O. May Lectures, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 21, hlm. 243–295, ISBN 978-0-387-25284-1
- ^ Guy, Richard (1994), Unsolved Problems in Number Theory (edisi ke-2nd), Springer, hlm. vii, ISBN 978-1-4899-3585-4, diarsipkan dari versi asli tanggal 2019-03-23, diakses tanggal 2016-09-22 .
- ^ Shimura, G. (1989). "Yutaka Taniyama and his time". Bulletin of the London Mathematical Society. 21 (2): 186–196. doi:10.1112/blms/21.2.186. Diarsipkan dari versi asli tanggal 2016-01-25. Diakses tanggal 2015-01-15.
- ^ "Archived copy" (PDF). Diarsipkan dari versi asli (PDF) tanggal 2016-02-08. Diakses tanggal 2016-01-22.
- ^ "THREE DIMENSIONAL MANIFOLDS, KLEINIAN GROUPS AND HYPERBOLIC GEOMETRY" (PDF). Diarsipkan dari versi asli (PDF) tanggal 2016-04-10. Diakses tanggal 2016-02-09.
- ^ "Millennium Problems". Diarsipkan dari versi asli tanggal 2017-06-06. Diakses tanggal 2015-01-20.
- ^ "Fields Medal awarded to Artur Avila". Centre national de la recherche scientifique. 2014-08-13. Diarsipkan dari versi asli tanggal 2018-07-10. Diakses tanggal 2018-07-07.
- ^ Bellos, Alex (2014-08-13). "Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained". The Guardian. Diarsipkan dari versi asli tanggal 2016-10-21. Diakses tanggal 2018-07-07.
- ^ Abe, Jair Minoro; Tanaka, Shotaro (2001). Unsolved Problems on Mathematics for the 21st Century. IOS Press. ISBN 978-9051994902.
- ^ "DARPA invests in math". CNN. 2008-10-14. Diarsipkan dari versi asli tanggal 2009-03-04. Diakses tanggal 2013-01-14.
- ^ "Broad Agency Announcement (BAA 07-68) for Defense Sciences Office (DSO)". DARPA. 2007-09-10. Diarsipkan dari versi asli tanggal 2012-10-01. Diakses tanggal 2013-06-25.
- ^ "Millennium Problems". Diarsipkan dari versi asli tanggal 2017-06-06. Diakses tanggal 2015-01-20.
- ^ "Poincaré Conjecture". Clay Mathematics Institute. Diarsipkan dari versi asli tanggal 2013-12-15.
- ^ "Smooth 4-dimensional Poincare conjecture". Diarsipkan dari versi asli tanggal 2018-01-25. Diakses tanggal 2019-08-06.
- ^ Dnestrovskaya notebook (PDF) (dalam bahasa Rusia), The Russian Academy of Sciences, 1993"Dneister Notebook: Unsolved Problems in the Theory of Rings and Modules" (PDF), University of Saskatchewan, diakses tanggal 2019-08-15
- ^ Erlagol notebook (PDF) (dalam bahasa Rusia), The Novosibirsk State University, 2018
- ^ Waldschmidt, Michel (2013), Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables, Springer, hlm. 14, 16, ISBN 9783662115695
- ^ Smyth, Chris (2008), "The Mahler measure of algebraic numbers: a survey", dalam McKee, James; Smyth, Chris, Number Theory and Polynomials, London Mathematical Society Lecture Note Series, 352, Cambridge University Press, hlm. 322–349, ISBN 978-0-521-71467-9
- ^ Berenstein, Carlos A. (2001) [1994], "Pompeiu problem", dalam Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- ^ Waldschmidt, Michel (2013), Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables, Springer, hlm. 14, 16, ISBN 9783662115695
- ^ For background on the numbers that are the focus of this problem, see articles by Eric W. Weisstein, on pi ( "Salinan arsip". Archived from the original on 2014-12-06. Diakses tanggal 2021-01-27. ), e ( "Salinan arsip". Archived from the original on 2014-11-21. Diakses tanggal 2021-01-27. ), Khinchin's Constant ( "Salinan arsip". Archived from the original on 2014-11-05. Diakses tanggal 2021-01-27. ), irrational numbers ( "Salinan arsip". Archived from the original on 2015-03-27. Diakses tanggal 2021-01-27. ), transcendental numbers ( "Salinan arsip". Archived from the original on 2014-11-13. Diakses tanggal 2021-01-27. ), and irrationality measures ( "Salinan arsip". Archived from the original on 2015-04-21. Diakses tanggal 2021-01-27. ) at Wolfram MathWorld, all articles accessed 15 December 2014.
- ^ Michel Waldschmidt, 2008, "An introduction to irrationality and transcendence methods," at The University of Arizona The Southwest Center for Arithmetic Geometry 2008 Arizona Winter School, March 15–19, 2008 (Special Functions and Transcendence), see "Salinan arsip" (PDF). Archived from the original on 2014-12-16. Diakses tanggal 2021-01-27. , accessed 15 December 2014.
- ^ John Albert, posting date unknown, "Some unsolved problems in number theory" [from Victor Klee & Stan Wagon, "Old and New Unsolved Problems in Plane Geometry and Number Theory"], in University of Oklahoma Math 4513 course materials, see "Salinan arsip" (PDF). Archived from the original on 2014-01-17. Diakses tanggal 2021-01-27. , accessed 15 December 2014.
- ^ Kung, H. T.; Traub, Joseph Frederick (1974), "Optimal order of one-point and multipoint iteration", Journal of the ACM, 21 (4): 643–651, doi:10.1145/321850.321860
- ^ Bruhn, Henning; Schaudt, Oliver (2015), "The journey of the union-closed sets conjecture" (PDF), Graphs and Combinatorics, 31 (6): 2043–2074, arXiv:1309.3297 , doi:10.1007/s00373-014-1515-0, MR 3417215, diarsipkan dari versi asli (PDF) tanggal 2017-08-08, diakses tanggal 2017-07-18
- ^ Tao, Terence (2017). "Some remarks on the lonely runner conjecture". arΧiv:1701.02048 [math.CO].
- ^ Liśkiewicz, Maciej; Ogihara, Mitsunori; Toda, Seinosuke (2003-07-28). "The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes". Theoretical Computer Science. 304 (1): 129–156. doi:10.1016/S0304-3975(03)00080-X.
- ^ Brightwell, Graham R.; Felsner, Stefan; Trotter, William T. (1995), "Balancing pairs and the cross product conjecture", Order, 12 (4): 327–349, CiteSeerX 10.1.1.38.7841 , doi:10.1007/BF01110378, MR 1368815 .
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