The Riemann zeta function, subject of the celebrated and influential unsolved problem known as the Riemann hypothesis

Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, yet many mathematical problems, both major and minor, still remain unsolved.[1] These unsolved problems occur in multiple domains, including physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph, group, model, number, set and Ramsey theories, dynamical systems, partial differential equations, and more. Some problems may belong to more than one discipline of mathematics and be studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems (such as the list of Millennium Prize Problems) receive considerable attention.

This article is a composite of notable unsolved problems derived from many sources, including but not limited to lists considered authoritative. It does not claim to be comprehensive, it may not always be quite up to date, and it includes problems which are considered by the mathematical community to be widely varying in both difficulty and centrality to the science as a whole.

## Lists of unsolved problems in mathematics

Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.

List Number of problems Number unresolved
or incompletely resolved
Proposed by Proposed in
Hilbert's problems[2] 23 15 David Hilbert 1900
Landau's problems[3] 4 4 Edmund Landau 1912
Taniyama's problems[4] 36 - Yutaka Taniyama 1955
Thurston's 24 questions[5][6] 24 - William Thurston 1982
Smale's problems 18 14 Stephen Smale 1998
Millennium Prize problems 7 6[7] Clay Mathematics Institute 2000
Simon problems 15 <12[8][9] Barry Simon 2000
Unsolved Problems on Mathematics for the 21st Century[10] 22 - Jair Minoro Abe, Shotaro Tanaka 2001
DARPA's math challenges[11][12] 23 - DARPA 2007

### Millennium Prize Problems

Of the original seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000, six have yet to be solved as of May, 2021:[7]

The seventh problem, the Poincaré conjecture, has been solved;[13] however, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is still unsolved.[14]

## Unsolved problems

### Algebra

In the Bloch sphere representation of a qubit, a SIC-POVM forms a regular tetrahedron. Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions.

#### Notebook problems

• The Dneister Notebook (Dnestrovskaya Tetrad) collects several hundred unresolved problems in algebra, particularly ring theory and modulus theory.[15]
• The Erlagol Notebook (Erlagolskaya Tetrad) collects unresolved problems in algebra and model theory.[16]

### Analysis

The area of the blue region converges to the Euler–Mascheroni constant, which may or may not be a rational number.

### Dynamical systems

A detail of the Mandelbrot set. It is not known whether the Mandelbrot set is locally connected or not.

### Geometry

#### Covering and packing

##### Conjectures and problems
• Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded n-dimensional set.
• The covering problem of Rado: if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?[38]
• The Erdős–Oler conjecture that when ${\displaystyle n}$ is a triangular number, packing ${\displaystyle n-1}$ circles in an equilateral triangle requires a triangle of the same size as packing ${\displaystyle n}$ circles[39]
• The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24[40]
• Reinhardt's conjecture that the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets[41]
• Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
• Square packing in a square: what is the asymptotic growth rate of wasted space?[42]
• Ulam's packing conjecture about the identity of the worst-packing convex solid[43]

#### Discrete geometry

In three dimensions, the kissing number is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a regular icosahedron.) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.

### Graph theory

#### Graph coloring and labeling

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.

### Group theory

The free Burnside group ${\displaystyle B(2,3)}$ is finite; in its Cayley graph, shown here, each of its 27 elements is represented by a vertex. The question of which other groups ${\displaystyle B(m,n)}$ are finite remains open.

#### Notebook problems

• The Kourovka Notebook is a collection of unsolved problems in group theory, first published in 1965 and updated many times since.[120]

### Model theory and formal languages

#### Conjectures and problems

• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in ${\displaystyle \aleph _{0))$ is a simple algebraic group over an algebraically closed field.
• Generalized star height problem
• For which number fields does Hilbert's tenth problem hold?
• Kueker's conjecture[122]
• The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for ${\displaystyle \aleph _{1))$-saturated models of a countable theory.[123]
• Shelah's categoricity conjecture for ${\displaystyle L_{\omega _{1},\omega ))$: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.[123]
• Shelah's eventual categoricity conjecture: For every cardinal ${\displaystyle \lambda }$ there exists a cardinal ${\displaystyle \mu (\lambda )}$ such that if an AEC K with LS(K)<= ${\displaystyle \lambda }$ is categorical in a cardinal above ${\displaystyle \mu (\lambda )}$ then it is categorical in all cardinals above ${\displaystyle \mu (\lambda )}$.[123][124]
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• The Stable Forking Conjecture for simple theories[125]
• Tarski's exponential function problem
• 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?[126]
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?[127]
• Vaught's conjecture

#### Open questions

• Is the theory of the field of Laurent series over ${\displaystyle \mathbb {Z} _{p))$ decidable? of the field of polynomials over ${\displaystyle \mathbb {C} }$?
• (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?[128]
• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality ${\displaystyle \aleph _{\omega _{1))}$ does it have a model of cardinality continuum?[129]
• If the class of atomic models of a complete first order theory is categorical in the ${\displaystyle \aleph _{n))$, is it categorical in every cardinal?[130][131]
• Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[132]
• 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.)
• Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
• Do the Henson graphs have the finite model property?

#### Other

• Determine the structure of Keisler's order[133][134]

### Number theory

#### General

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.

#### Algebraic number theory

##### Other
• Characterize all algebraic number fields that have some power basis.

#### Prime numbers

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.

### Topology

The unknotting problem asks whether there is an efficient algorithm to identify when the shape presented in a knot diagram is actually the unknot.

## Problems solved since 1995

Ricci flow, here illustrated with a 2D manifold, was the key tool in Grigori Perelman's solution of the Poincaré conjecture.

## References

1. ^ Eves, An Introduction to the History of Mathematics 6th Edition, Thomson, 1990, ISBN 978-0-03-029558-4.
2. ^ Thiele, Rüdiger (2005), "On Hilbert and his twenty-four problems", in Van Brummelen, Glen (ed.), Mathematics and the historian's craft. The Kenneth O. May Lectures, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 21, pp. 243–295, ISBN 978-0-387-25284-1
3. ^ Guy, Richard (1994), Unsolved Problems in Number Theory (2nd ed.), Springer, p. vii, ISBN 978-1-4899-3585-4, archived from the original on 2019-03-23, retrieved 2016-09-22.
4. ^ 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.
5. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2016-02-08. Retrieved 2016-01-22.CS1 maint: archived copy as title (link)
6. ^ "THREE DIMENSIONAL MANIFOLDS, KLEINIAN GROUPS AND HYPERBOLIC GEOMETRY" (PDF). Archived (PDF) from the original on 2016-04-10. Retrieved 2016-02-09.
7. ^ a b "Millennium Problems". Archived from the original on 2017-06-06. Retrieved 2015-01-20.
8. ^ "Fields Medal awarded to Artur Avila". Centre national de la recherche scientifique. 2014-08-13. Archived from the original on 2018-07-10. Retrieved 2018-07-07.
9. ^ Bellos, Alex (2014-08-13). "Fields Medals 2014: the maths of Avila, Bhargava, Hairer and Mirzakhani explained". The Guardian. Archived from the original on 2016-10-21. Retrieved 2018-07-07.
10. ^ Abe, Jair Minoro; Tanaka, Shotaro (2001). Unsolved Problems on Mathematics for the 21st Century. IOS Press. ISBN 978-9051994902.
11. ^ "DARPA invests in math". CNN. 2008-10-14. Archived from the original on 2009-03-04. Retrieved 2013-01-14.
12. ^ "Broad Agency Announcement (BAA 07-68) for Defense Sciences Office (DSO)". DARPA. 2007-09-10. Archived from the original on 2012-10-01. Retrieved 2013-06-25.
13. ^ "Poincaré Conjecture". Clay Mathematics Institute. Archived from the original on 2013-12-15.
14. ^ "Smooth 4-dimensional Poincare conjecture". Archived from the original on 2018-01-25. Retrieved 2019-08-06.
15. ^ Dnestrovskaya notebook (PDF) (in Russian), The Russian Academy of Sciences, 1993
"Dneister Notebook: Unsolved Problems in the Theory of Rings and Modules" (PDF), University of Saskatchewan, retrieved 2019-08-15
16. ^ Erlagol notebook (PDF) (in Russian), The Novosibirsk State University, 2018
17. ^ a b Waldschmidt, Michel (2013), Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential Function in Several Variables, Springer, pp. 14, 16, ISBN 9783662115695
18. ^ Smyth, Chris (2008), "The Mahler measure of algebraic numbers: a survey", in McKee, James; Smyth, Chris (eds.), Number Theory and Polynomials, London Mathematical Society Lecture Note Series, 352, Cambridge University Press, pp. 322–349, ISBN 978-0-521-71467-9
19. ^ Berenstein, Carlos A. (2001) [1994], "Pompeiu problem", Encyclopedia of Mathematics, EMS Press
20. ^ 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, S2CID 74921
21. ^ For background on the numbers that are the focus of this problem, see articles by Eric W. Weisstein, on pi ([1] Archived 2014-12-06 at the Wayback Machine), e ([2] Archived 2014-11-21 at the Wayback Machine), Khinchin's Constant ([3] Archived 2014-11-05 at the Wayback Machine), irrational numbers ([4] Archived 2015-03-27 at the Wayback Machine), transcendental numbers ([5] Archived 2014-11-13 at the Wayback Machine), and irrationality measures ([6] Archived 2015-04-21 at the Wayback Machine) at Wolfram MathWorld, all articles accessed 15 December 2014.
22. ^ 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 [7] Archived 2014-12-16 at the Wayback Machine, accessed 15 December 2014.
23. ^ 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 [8] Archived 2014-01-17 at the Wayback Machine, accessed 15 December 2014.
24. ^ 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, S2CID 14793475.
25. ^ Tao, Terence (2017), "Some remarks on the lonely runner conjecture", arXiv:1701.02048 [math.CO]
26. ^ 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, S2CID 17531822, archived (PDF) from the original on 2017-08-08, retrieved 2017-07-18
27. ^ 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.
28. ^ Dedekind Numbers and Related Sequences
29. ^ Murnaghan, F. D. (1938), "The Analysis of the Direct Product of Irreducible Representations of the Symmetric Groups", American Journal of Mathematics, 60 (1): 44–65, doi:10.2307/2371542, JSTOR 2371542, MR 1507301, PMC 1076971, PMID 16577800
30. ^ Kaloshin, Vadim; Sorrentino, Alfonso (2018). "On the local Birkhoff conjecture for convex billiards". Annals of Mathematics. 188 (1): 315–380. arXiv:1612.09194. doi:10.4007/annals.2018.188.1.6. S2CID 119171182.
31. ^ Sarnak, Peter (2011), "Recent progress on the quantum unique ergodicity conjecture", Bulletin of the American Mathematical Society, 48 (2): 211–228, doi:10.1090/S0273-0979-2011-01323-4, MR 2774090
32. ^ Kari, Jarkko (2009), "Structure of reversible cellular automata", Unconventional Computation: 8th International Conference, UC 2009, Ponta Delgada, Portugal, September 7ÔÇô11, 2009, Proceedings, Lecture Notes in Computer Science, 5715, Springer, p. 6, Bibcode:2009LNCS.5715....6K, doi:10.1007/978-3-642-03745-0_5, ISBN 978-3-642-03744-3
33. ^ a b c http://english.log-it-ex.com Archived 2017-11-10 at the Wayback Machine Ten open questions about Sudoku (2012-01-21).
34. ^ "Higher-Dimensional Tic-Tac-Toe". PBS Infinite Series. YouTube. 2017-09-21. Archived from the original on 2017-10-11. Retrieved 2018-07-29.
35. ^ Barlet, Daniel; Peternell, Thomas; Schneider, Michael (1990). "On two conjectures of Hartshorne's". Mathematische Annalen. 286 (1–3): 13–25. doi:10.1007/BF01453563. S2CID 122151259.
36. ^ Maulik, Davesh; Nekrasov, Nikita; Okounov, Andrei; Pandharipande, Rahul (2004-06-05), Gromov–Witten theory and Donaldson–Thomas theory, I, arXiv:math/0312059, Bibcode:2003math.....12059M
37. ^
38. ^ Bereg, Sergey; Dumitrescu, Adrian; Jiang, Minghui (2010), "On covering problems of Rado", Algorithmica, 57 (3): 538–561, doi:10.1007/s00453-009-9298-z, MR 2609053, S2CID 6511998
39. ^ Melissen, Hans (1993), "Densest packings of congruent circles in an equilateral triangle", American Mathematical Monthly, 100 (10): 916–925, doi:10.2307/2324212, JSTOR 2324212, MR 1252928
40. ^ Conway, John H.; Neil J.A. Sloane (1999), Sphere Packings, Lattices and Groups (3rd ed.), New York: Springer-Verlag, pp. 21–22, ISBN 978-0-387-98585-5
41. ^ Hales, Thomas (2017), The Reinhardt conjecture as an optimal control problem, arXiv:1703.01352
42. ^ Brass, Peter; Moser, William; Pach, János (2005), Research Problems in Discrete Geometry, New York: Springer, p. 45, ISBN 978-0387-23815-9, MR 2163782
43. ^ Gardner, Martin (1995), New Mathematical Diversions (Revised Edition), Washington: Mathematical Association of America, p. 251
44. ^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", Proceedings of the American Mathematical Society, 125 (5): 1503–1509, doi:10.1090/S0002-9939-97-03692-7, JSTOR 2162098
45. ^ Katz, Mikhail G. (2007), Systolic geometry and topology, Mathematical Surveys and Monographs, 137, American Mathematical Society, Providence, RI, p. 57, doi:10.1090/surv/137, ISBN 978-0-8218-4177-8, MR 2292367
46. ^ Rosenberg, Steven (1997), The Laplacian on a Riemannian Manifold: An introduction to analysis on manifolds, London Mathematical Society Student Texts, 31, Cambridge: Cambridge University Press, pp. 62–63, doi:10.1017/CBO9780511623783, ISBN 978-0-521-46300-3, MR 1462892
47. ^ Boltjansky, V.; Gohberg, I. (1985), "11. Hadwiger's Conjecture", Results and Problems in Combinatorial Geometry, Cambridge University Press, pp. 44–46.
48. ^ Morris, Walter D.; Soltan, Valeriu (2000), "The Erdős-Szekeres problem on points in convex position—a survey", Bull. Amer. Math. Soc., 37 (4): 437–458, doi:10.1090/S0273-0979-00-00877-6, MR 1779413; Suk, Andrew (2016), "On the Erdős–Szekeres convex polygon problem", J. Amer. Math. Soc., 30 (4): 1047–1053, arXiv:1604.08657, doi:10.1090/jams/869, S2CID 15732134
49. ^ Kalai, Gil (1989), "The number of faces of centrally-symmetric polytopes", Graphs and Combinatorics, 5 (1): 389–391, doi:10.1007/BF01788696, MR 1554357, S2CID 8917264.
50. ^
51. ^ Guy, Richard K. (1983), "An olla-podrida of open problems, often oddly posed", American Mathematical Monthly, 90 (3): 196–200, doi:10.2307/2975549, JSTOR 2975549, MR 1540158
52. ^ Matoušek, Jiří (2002), Lectures on discrete geometry, Graduate Texts in Mathematics, 212, Springer-Verlag, New York, p. 206, doi:10.1007/978-1-4613-0039-7, ISBN 978-0-387-95373-1, MR 1899299
53. ^ Brass, Peter; Moser, William; Pach, János (2005), "5.1 The Maximum Number of Unit Distances in the Plane", Research problems in discrete geometry, Springer, New York, pp. 183–190, ISBN 978-0-387-23815-9, MR 2163782
54. ^ Dey, Tamal K. (1998), "Improved bounds for planar k-sets and related problems", Discrete Comput. Geom., 19 (3): 373–382, doi:10.1007/PL00009354, MR 1608878; Tóth, Gábor (2001), "Point sets with many k-sets", Discrete Comput. Geom., 26 (2): 187–194, doi:10.1007/s004540010022, MR 1843435.
55. ^ Aronov, Boris; Dujmović, Vida; Morin, Pat; Ooms, Aurélien; Schultz Xavier da Silveira, Luís Fernando (2019), "More Turán-type theorems for triangles in convex point sets", Electronic Journal of Combinatorics, 26 (1): P1.8, arXiv:1706.10193, Bibcode:2017arXiv170610193A, doi:10.37236/7224, archived from the original on 2019-02-18, retrieved 2019-02-18
56. ^ Atiyah, Michael (2001), "Configurations of points", Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 359 (1784): 1375–1387, Bibcode:2001RSPTA.359.1375A, doi:10.1098/rsta.2001.0840, ISSN 1364-503X, MR 1853626, S2CID 55833332
57. ^ Finch, S. R.; Wetzel, J. E. (2004), "Lost in a forest", American Mathematical Monthly, 11 (8): 645–654, doi:10.2307/4145038, JSTOR 4145038, MR 2091541
58. ^ Solomon, Yaar; Weiss, Barak (2016), "Dense forests and Danzer sets", Annales Scientifiques de l'École Normale Supérieure, 49 (5): 1053–1074, arXiv:1406.3807, doi:10.24033/asens.2303, MR 3581810, S2CID 672315; Conway, John H., Five $1,000 Problems (Update 2017) (PDF), On-Line Encyclopedia of Integer Sequences, archived (PDF) from the original on 2019-02-13, retrieved 2019-02-12 59. ^ Socolar, Joshua E. S.; Taylor, Joan M. (2012), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer, 34 (1): 18–28, arXiv:1009.1419, doi:10.1007/s00283-011-9255-y, MR 2902144, S2CID 10747746 60. ^ Arutyunyants, G.; Iosevich, A. (2004), "Falconer conjecture, spherical averages and discrete analogs", in Pach, János (ed.), Towards a Theory of Geometric Graphs, Contemp. Math., 342, Amer. Math. Soc., Providence, RI, pp. 15–24, doi:10.1090/conm/342/06127, ISBN 9780821834848, MR 2065249 61. ^ Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society, 61 (4): 346–352, doi:10.1090/noti1100 62. ^ Katz, Nets; Tao, Terence (2002), "Recent progress on the Kakeya conjecture", Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), Publicacions Matemàtiques (Vol. Extra): 161–179, CiteSeerX 10.1.1.241.5335, doi:10.5565/PUBLMAT_Esco02_07, MR 1964819, S2CID 77088 63. ^ Weaire, Denis, ed. (1997), The Kelvin Problem, CRC Press, p. 1, ISBN 9780748406326 64. ^ Brass, Peter; Moser, William; Pach, János (2005), Research problems in discrete geometry, New York: Springer, p. 457, ISBN 9780387299297, MR 2163782 65. ^ Mahler, Kurt (1939). "Ein Minimalproblem für konvexe Polygone". Mathematica (Zutphen) B: 118–127. 66. ^ Norwood, Rick; Poole, George; Laidacker, Michael (1992), "The worm problem of Leo Moser", Discrete and Computational Geometry, 7 (2): 153–162, doi:10.1007/BF02187832, MR 1139077 67. ^ Wagner, Neal R. (1976), "The Sofa Problem" (PDF), The American Mathematical Monthly, 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022, archived (PDF) from the original on 2015-04-20, retrieved 2014-05-14 68. ^ Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338 69. ^ Ghomi, Mohammad (2018-01-01). "D "urer's Unfolding Problem for Convex Polyhedra". Notices of the American Mathematical Society. 65 (1): 25–27. doi:10.1090/noti1609. ISSN 0002-9920. 70. ^ Whyte, L. L. (1952), "Unique arrangements of points on a sphere", The American Mathematical Monthly, 59 (9): 606–611, doi:10.2307/2306764, JSTOR 2306764, MR 0050303 71. ^ Howards, Hugh Nelson (2013), "Forming the Borromean rings out of arbitrary polygonal unknots", Journal of Knot Theory and Its Ramifications, 22 (14): 1350083, 15, arXiv:1406.3370, doi:10.1142/S0218216513500831, MR 3190121, S2CID 119674622 72. ^ Brandts, Jan; Korotov, Sergey; Křížek, Michal; Šolc, Jakub (2009), "On nonobtuse simplicial partitions" (PDF), SIAM Review, 51 (2): 317–335, Bibcode:2009SIAMR..51..317B, doi:10.1137/060669073, MR 2505583, archived (PDF) from the original on 2018-11-04, retrieved 2018-11-22. See in particular Conjecture 23, p. 327. 73. ^ ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, archived from the original on October 5, 2016, retrieved 2016-10-04. 74. ^ Bousquet, Nicolas; Bartier, Valentin (2019), "Linear Transformations Between Colorings in Chordal Graphs", in Bender, Michael A.; Svensson, Ola; Herman, Grzegorz (eds.), 27th Annual European Symposium on Algorithms, ESA 2019, September 9-11, 2019, Munich/Garching, Germany, LIPIcs, 144, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 24:1–24:15, doi:10.4230/LIPIcs.ESA.2019.24, S2CID 195791634 75. ^ Chung, Fan; Graham, Ron (1998), Erdős on Graphs: His Legacy of Unsolved Problems, A K Peters, pp. 97–99. 76. ^ Chudnovsky, Maria; Seymour, Paul (2014), "Extending the Gyárfás-Sumner conjecture", Journal of Combinatorial Theory, Series B, 105: 11–16, doi:10.1016/j.jctb.2013.11.002, MR 3171779 77. ^ Toft, Bjarne (1996), "A survey of Hadwiger's conjecture", Congressus Numerantium, 115: 249–283, MR 1411244. 78. ^ Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1991), Unsolved Problems in Geometry, Springer-Verlag, Problem G10. 79. ^ Hägglund, Jonas; Steffen, Eckhard (2014), "Petersen-colorings and some families of snarks", Ars Mathematica Contemporanea, 7 (1): 161–173, doi:10.26493/1855-3974.288.11a, MR 3047618, archived from the original on 2016-10-03, retrieved 2016-09-30. 80. ^ Jensen, Tommy R.; Toft, Bjarne (1995), "12.20 List-Edge-Chromatic Numbers", Graph Coloring Problems, New York: Wiley-Interscience, pp. 201–202, ISBN 978-0-471-02865-9. 81. ^ Molloy, Michael; Reed, Bruce (1998), "A bound on the total chromatic number", Combinatorica, 18 (2): 241–280, CiteSeerX 10.1.1.24.6514, doi:10.1007/PL00009820, MR 1656544, S2CID 9600550. 82. ^ Barát, János; Tóth, Géza (2010), "Towards the Albertson Conjecture", Electronic Journal of Combinatorics, 17 (1): R73, arXiv:0909.0413, Bibcode:2009arXiv0909.0413B, doi:10.37236/345. 83. ^ Wood, David (January 19, 2009), "Book Thickness of Subdivisions", Open Problem Garden, archived from the original on September 16, 2013, retrieved 2013-02-05. 84. ^ Fulek, R.; Pach, J. (2011), "A computational approach to Conway's thrackle conjecture", Computational Geometry, 44 (6–7): 345–355, arXiv:1002.3904, doi:10.1007/978-3-642-18469-7_21, MR 2785903. 85. ^ Hartsfield, Nora; Ringel, Gerhard (2013), Pearls in Graph Theory: A Comprehensive Introduction, Dover Books on Mathematics, Courier Dover Publications, p. 247, ISBN 978-0-486-31552-2, MR 2047103. 86. ^ Hliněný, Petr (2010), "20 years of Negami's planar cover conjecture" (PDF), Graphs and Combinatorics, 26 (4): 525–536, CiteSeerX 10.1.1.605.4932, doi:10.1007/s00373-010-0934-9, MR 2669457, S2CID 121645, archived (PDF) from the original on 2016-03-04, retrieved 2016-10-04. 87. ^ Nöllenburg, Martin; Prutkin, Roman; Rutter, Ignaz (2016), "On self-approaching and increasing-chord drawings of 3-connected planar graphs", Journal of Computational Geometry, 7 (1): 47–69, arXiv:1409.0315, doi:10.20382/jocg.v7i1a3, MR 3463906 88. ^ Pach, János; Sharir, Micha (2009), "5.1 Crossings—the Brick Factory Problem", Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures, Mathematical Surveys and Monographs, 152, American Mathematical Society, pp. 126–127. 89. ^ Demaine, E.; O'Rourke, J. (2002–2012), "Problem 45: Smallest Universal Set of Points for Planar Graphs", The Open Problems Project, archived from the original on 2012-08-14, retrieved 2013-03-19. 90. ^ Florek, Jan (2010), "On Barnette's conjecture", Discrete Mathematics, 310 (10–11): 1531–1535, doi:10.1016/j.disc.2010.01.018, MR 2601261. 91. ^ Broersma, Hajo; Patel, Viresh; Pyatkin, Artem (2014), "On toughness and Hamiltonicity of$2K_2$-free graphs", Journal of Graph Theory, 75 (3): 244–255, doi:10.1002/jgt.21734, MR 3153119 92. ^ Jaeger, F. (1985), "A survey of the cycle double cover conjecture", Annals of Discrete Mathematics 27 – Cycles in Graphs, North-Holland Mathematics Studies, 27, pp. 1–12, doi:10.1016/S0304-0208(08)72993-1, ISBN 9780444878038. 93. ^ Heckman, Christopher Carl; Krakovski, Roi (2013), "Erdös-Gyárfás conjecture for cubic planar graphs", Electronic Journal of Combinatorics, 20 (2), P7, doi:10.37236/3252. 94. ^ Akiyama, Jin; Exoo, Geoffrey; Harary, Frank (1981), "Covering and packing in graphs. IV. Linear arboricity", Networks, 11 (1): 69–72, doi:10.1002/net.3230110108, MR 0608921. 95. ^ L. Babai, Automorphism groups, isomorphism, reconstruction Archived 2007-06-13 at the Wayback Machine, in Handbook of Combinatorics, Vol. 2, Elsevier, 1996, 1447–1540. 96. ^ Lenz, Hanfried; Ringel, Gerhard (1991), "A brief review on Egmont Köhler's mathematical work", Discrete Mathematics, 97 (1–3): 3–16, doi:10.1016/0012-365X(91)90416-Y, MR 1140782 97. S. Kitaev and V. Lozin. Words and Graphs, Springer, 2015. 98. S. Kitaev and A. Pyatkin. Word-representable graphs: a Survey, Journal of Applied and Industrial Mathematics 12(2) (2018) 278−296. 99. С. В. Китаев, А. В. Пяткин. Графы, представимые в виде слов. Обзор результатов, Дискретн. анализ и исслед. опер., 2018, том 25,номер 2, 19−53 100. ^ Marc Elliot Glen (2016). "Colourability and word-representability of near-triangulations". arXiv:1605.01688 [math.CO]. 101. ^ S. Kitaev. On graphs with representation number 3, J. Autom., Lang. and Combin. 18 (2013), 97−112. 102. ^ Glen, Marc; Kitaev, Sergey; Pyatkin, Artem (2018). "On the representation number of a crown graph". Discrete Applied Mathematics. 244: 89–93. arXiv:1609.00674. doi:10.1016/j.dam.2018.03.013. S2CID 46925617. 103. ^ Conway, John H., Five$1,000 Problems (Update 2017) (PDF), Online Encyclopedia of Integer Sequences, archived (PDF) from the original on 2019-02-13, retrieved 2019-02-12
104. ^ Chudnovsky, Maria (2014), "The Erdös–Hajnal conjecture—a survey" (PDF), Journal of Graph Theory, 75 (2): 178–190, arXiv:1606.08827, doi:10.1002/jgt.21730, MR 3150572, S2CID 985458, Zbl 1280.05086, archived (PDF) from the original on 2016-03-04, retrieved 2016-09-22.
105. ^ Gupta, Anupam; Newman, Ilan; Rabinovich, Yuri; Sinclair, Alistair (2004), "Cuts, trees and ${\displaystyle \ell _{1))$-embeddings of graphs", Combinatorica, 24 (2): 233–269, CiteSeerX 10.1.1.698.8978, doi:10.1007/s00493-004-0015-x, MR 2071334, S2CID 46133408
106. ^ Pleanmani, Nopparat (2019), "Graham's pebbling conjecture holds for the product of a graph and a sufficiently large complete bipartite graph", Discrete Mathematics, Algorithms and Applications, 11 (6): 1950068, 7, doi:10.1142/s179383091950068x, MR 4044549
107. ^ Spinrad, Jeremy P. (2003), "2. Implicit graph representation", Efficient Graph Representations, pp. 17–30, ISBN 978-0-8218-2815-1.
108. ^ "Jorgensen's Conjecture", Open Problem Garden, archived from the original on 2016-11-14, retrieved 2016-11-13.
109. ^ Baird, William; Bonato, Anthony (2012), "Meyniel's conjecture on the cop number: a survey", Journal of Combinatorics, 3 (2): 225–238, arXiv:1308.3385, doi:10.4310/JOC.2012.v3.n2.a6, MR 2980752, S2CID 18942362
110. ^ Schwenk, Allen (2012), "Some History on the Reconstruction Conjecture" (PDF), Joint Mathematics Meetings, archived (PDF) from the original on 2015-04-09, retrieved 2018-11-26
111. ^ Ramachandran, S. (1981), "On a new digraph reconstruction conjecture", Journal of Combinatorial Theory, Series B, 31 (2): 143–149, doi:10.1016/S0095-8956(81)80019-6, MR 0630977
112. ^ Seymour's 2nd Neighborhood Conjecture Archived 2019-01-11 at the Wayback Machine, Open Problems in Graph Theory and Combinatorics, Douglas B. West.
113. ^ Blokhuis, A.; Brouwer, A. E. (1988), "Geodetic graphs of diameter two", Geometriae Dedicata, 25 (1–3): 527–533, doi:10.1007/BF00191941, MR 0925851, S2CID 189890651
114. ^ Kühn, Daniela; Mycroft, Richard; Osthus, Deryk (2011), "A proof of Sumner's universal tournament conjecture for large tournaments", Proceedings of the London Mathematical Society, Third Series, 102 (4): 731–766, arXiv:1010.4430, doi:10.1112/plms/pdq035, MR 2793448, S2CID 119169562, Zbl 1218.05034.
115. ^ 4-flow conjecture Archived 2018-11-26 at the Wayback Machine and 5-flow conjecture Archived 2018-11-26 at the Wayback Machine, Open Problem Garden
116. ^ Brešar, Boštjan; Dorbec, Paul; Goddard, Wayne; Hartnell, Bert L.; Henning, Michael A.; Klavžar, Sandi; Rall, Douglas F. (2012), "Vizing's conjecture: a survey and recent results", Journal of Graph Theory, 69 (1): 46–76, CiteSeerX 10.1.1.159.7029, doi:10.1002/jgt.20565, MR 2864622.
117. ^ Ducey, Joshua E. (2017), "On the critical group of the missing Moore graph", Discrete Mathematics, 340 (5): 1104–1109, arXiv:1509.00327, doi:10.1016/j.disc.2016.10.001, MR 3612450, S2CID 28297244
118. ^ Fomin, Fedor V.; Høie, Kjartan (2006), "Pathwidth of cubic graphs and exact algorithms", Information Processing Letters, 97 (5): 191–196, doi:10.1016/j.ipl.2005.10.012, MR 2195217
119. ^ Khukhro, Evgeny I.; Mazurov, Victor D. (2019), Unsolved Problems in Group Theory. The Kourovka Notebook, arXiv:1401.0300v16
120. ^ Aschbacher, Michael (1990), "On Conjectures of Guralnick and Thompson", Journal of Algebra, 135 (2): 277–343, doi:10.1016/0021-8693(90)90292-V
121. ^ Hrushovski, Ehud (1989). "Kueker's conjecture for stable theories". Journal of Symbolic Logic. 54 (1): 207–220. doi:10.2307/2275025. JSTOR 2275025.
122. ^ a b c Shelah S, Classification Theory, North-Holland, 1990
123. ^ Shelah, Saharon (2009). Classification theory for abstract elementary classes. College Publications. ISBN 978-1-904987-71-0.
124. ^ Peretz, Assaf (2006). "Geometry of forking in simple theories". Journal of Symbolic Logic. 71 (1): 347–359. arXiv:math/0412356. doi:10.2178/jsl/1140641179. S2CID 9380215.
125. ^ Cherlin, G.; Shelah, S. (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory, Series B. 97 (3): 293–333. arXiv:math/0512218. doi:10.1016/j.jctb.2006.05.008. S2CID 10425739.
126. ^ Džamonja, Mirna, "Club guessing and the universal models." On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
127. ^ Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
128. ^ Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae. 159 (1): 1–50. arXiv:math/9802134. Bibcode:1998math......2134S. doi:10.4064/fm-159-1-1-50. S2CID 8846429.
129. ^ Baldwin, John T. (July 24, 2009). Categoricity (PDF). American Mathematical Society. ISBN 978-0-8218-4893-7. Archived (PDF) from the original on July 29, 2010. Retrieved February 20, 2014.
130. ^ Shelah, Saharon (2009). "Introduction to classification theory for abstract elementary classes". arXiv:0903.3428. Bibcode:2009arXiv0903.3428S. Cite journal requires |journal= (help)
131. ^ Makowsky J, "Compactness, embeddings and definability," in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
132. ^ Keisler, HJ (1967). "Ultraproducts which are not saturated". J. Symb. Log. 32 (1): 23–46. doi:10.2307/2271240. JSTOR 2271240.
133. ^ Malliaris M, Shelah S, "A dividing line in simple unstable theories." https://arxiv.org/abs/1208.2140 Archived 2017-08-02 at the Wayback Machine
134. ^ Conrey, Brian (2016), "Lectures on the Riemann zeta function (book review)", Bulletin of the American Mathematical Society, 53 (3): 507–512, doi:10.1090/bull/1525
135. ^ Singmaster, D. (1971), "Research Problems: How often does an integer occur as a binomial coefficient?", American Mathematical Monthly, 78 (4): 385–386, doi:10.2307/2316907, JSTOR 2316907, MR 1536288.
136. ^ Aigner, Martin (2013), Markov's theorem and 100 years of the uniqueness conjecture, Cham: Springer, doi:10.1007/978-3-319-00888-2, ISBN 978-3-319-00887-5, MR 3098784
137. ^ "Are the Digits of Pi Random? Berkeley Lab Researcher May Hold Key". Archived from the original on 2016-03-27. Retrieved 2016-03-18.
138. ^ Bruhn, Henning; Schaudt, Oliver (2016). "Newer sums of three cubes". arXiv:1604.07746v1 [math.NT].
139. ^ Guo, Song; Sun, Zhi-Wei (2005), "On odd covering systems with distinct moduli", Advances in Applied Mathematics, 35 (2): 182–187, arXiv:math/0412217, doi:10.1016/j.aam.2005.01.004, MR 2152886, S2CID 835158
140. ^ Ribenboim, P. (2006). Die Welt der Primzahlen. Springer-Lehrbuch (in German) (2nd ed.). Springer. pp. 242–243. doi:10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
141. ^ Dobson, J. B. (1 April 2017), "On Lerch's formula for the Fermat quotient", p. 23, arXiv:1103.3907v6 [math.NT]
142. ^ Mazur, Barry (1992), "The topology of rational points", Experimental Mathematics, 1 (1): 35–45, doi:10.1080/10586458.1992.10504244 (inactive 31 May 2021), archived from the original on 2019-04-07, retrieved 2019-04-07CS1 maint: DOI inactive as of May 2021 (link)
143. ^ Casazza, Peter G.; Fickus, Matthew; Tremain, Janet C.; Weber, Eric (2006). "The Kadison-Singer problem in mathematics and engineering: A detailed account". In Han, Deguang; Jorgensen, Palle E. T.; Larson, David Royal (eds.). Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7–12, 2005, University of Central Florida, Florida. Contemporary Mathematics. 414. American Mathematical Society. pp. 299–355. doi:10.1090/conm/414/07820. ISBN 978-0-8218-3923-2. Retrieved 24 April 2015.
144. ^ Mackenzie, Dana. "Kadison–Singer Problem Solved" (PDF). SIAM News (January/February 2014). Society for Industrial and Applied Mathematics. Archived (PDF) from the original on 23 October 2014. Retrieved 24 April 2015.
145. ^ Moreira, Joel; Richter, Florian K.; Robertson, Donald (2019). "A proof of a sumset conjecture of Erdős". Annals of Mathematics. 189 (2): 605–652. arXiv:1803.00498. doi:10.4007/annals.2019.189.2.4. S2CID 119158401.
146. ^ Stanley, Richard P. (1994), "A survey of Eulerian posets", in Bisztriczky, T.; McMullen, P.; Schneider, R.; Weiss, A. IviÄ‡ (eds.), Polytopes: abstract, convex and computational (Scarborough, ON, 1993), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 440, Dordrecht: Kluwer Academic Publishers, pp. 301–333, MR 1322068. See in particular p. 316.
147. ^ Kalai, Gil (2018-12-25). "Amazing: Karim Adiprasito proved the g-conjecture for spheres!". Archived from the original on 2019-02-16. Retrieved 2019-02-15.
148. ^ Santos, Franciscos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics. 176 (1): 383–412. arXiv:1006.2814. doi:10.4007/annals.2012.176.1.7. S2CID 15325169.
149. ^ Ziegler, Günter M. (2012). "Who solved the Hirsch conjecture?". Documenta Mathematica. Extra Volume "Optimization Stories": 75–85. Archived from the original on 2015-04-02. Retrieved 2015-03-25.
150. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
151. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2016-01-07. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
152. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-10-13. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
153. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
154. ^
155. ^ Wolchover, Natalie (July 11, 2017), "Pentagon Tiling Proof Solves Century-Old Math Problem", Quanta Magazine, archived from the original on August 6, 2017, retrieved July 18, 2017
156. ^ Bruhn, Henning; Schaudt, Oliver (2010). "On the Erdos distinct distance problem in the plane". arXiv:1011.4105v3 [math.CO].
157. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-03-24. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
158. ^ Bruhn, Henning; Schaudt, Oliver (2015). "A formal proof of the Kepler conjecture". arXiv:1501.02155 [math.MG].
159. ^ Bruhn, Henning; Schaudt, Oliver (1998). "A proof of the dodecahedral conjecture". arXiv:math/9811079.
160. ^ Huang, C.; Kotzig, A.; Rosa, A. (1982), "Further results on tree labellings", Utilitas Mathematica, 21: 31–48, MR 0668845.
161. ^ Hartnett, Kevin (19 February 2020). "Rainbow Proof Shows Graphs Have Uniform Parts". Quanta Magazine. Retrieved 2020-02-29.
162. ^ Shitov, Yaroslav (May 2019). "Counterexamples to Hedetniemi's conjecture". arXiv:1905.02167 [math.CO].
163. ^ Abdollahi A., Zallaghi M. (2015). "Character sums for Cayley graphs". Communications in Algebra. 43 (12): 5159–5167. doi:10.1080/00927872.2014.967398. S2CID 117651702.
164. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2016-03-03. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
165. ^ Bruhn, Henning; Schaudt, Oliver (2005). "Menger's theorem for infinite graphs". arXiv:math/0509397.
166. ^ Seigel-Itzkovich, Judy (2008-02-08). "Russian immigrant solves math puzzle". The Jerusalem Post. Retrieved 2015-11-12.
167. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-10-07. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
168. ^ Namazi, Hossein; Souto, Juan (2012). "Non-realizability and ending laminations: Proof of the density conjecture". Acta Mathematica. 209 (2): 323–395. doi:10.1007/s11511-012-0088-0.
169. ^ Bourgain, Jean; Ciprian, Demeter; Larry, Guth (2015). "Proof of the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three". Annals of Mathematics. 184 (2): 633–682. arXiv:1512.01565. Bibcode:2015arXiv151201565B. doi:10.4007/annals.2016.184.2.7. hdl:1721.1/115568. S2CID 43929329.
170. ^ Helfgott, Harald A. (2013). "Major arcs for Goldbach's theorem". arXiv:1305.2897 [math.NT].
171. ^ Helfgott, Harald A. (2012). "Minor arcs for Goldbach's problem". arXiv:1205.5252 [math.NT].
172. ^ Helfgott, Harald A. (2013). "The ternary Goldbach conjecture is true". arXiv:1312.7748 [math.NT].
173. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)", Inventiones Mathematicae, 178 (3): 485–504, Bibcode:2009InMat.178..485K, CiteSeerX 10.1.1.518.4611, doi:10.1007/s00222-009-0205-7, S2CID 14846347
174. ^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)", Inventiones Mathematicae, 178 (3): 505–586, Bibcode:2009InMat.178..505K, CiteSeerX 10.1.1.228.8022, doi:10.1007/s00222-009-0206-6, S2CID 189820189
175. ^ "2011 Cole Prize in Number Theory" (PDF). Notices of the AMS. 58 (4): 610–611. ISSN 1088-9477. OCLC 34550461. Archived (PDF) from the original on 2015-11-06. Retrieved 2015-11-12.
176. ^ Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255. Archived (PDF) from the original on 2011-05-10. Retrieved 2016-03-06.
177. ^ Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. 141 (3): 553–572. CiteSeerX 10.1.1.128.531. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255.
178. ^ Lee, Choongbum (2017). "Ramsey numbers of degenerate graphs". Annals of Mathematics. 185 (3): 791–829. arXiv:1505.04773. doi:10.4007/annals.2017.185.3.2. S2CID 7974973.
179. ^ Lamb, Evelyn (26 May 2016). "Two-hundred-terabyte maths proof is largest ever". Nature. 534 (7605): 17–18. Bibcode:2016Natur.534...17L. doi:10.1038/nature.2016.19990. PMID 27251254.
180. ^ Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016). "Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer". In Creignou, N.; Le Berre, D. (eds.). Theory and Applications of Satisfiability Testing – SAT 2016. Lecture Notes in Computer Science. 9710. Springer, [Cham]. pp. 228–245. arXiv:1605.00723. doi:10.1007/978-3-319-40970-2_15. ISBN 978-3-319-40969-6. MR 3534782. S2CID 7912943.
181. ^ The Conway knot is not slice, Annals of Mathematics, volume 191, issue 2, pp. 581–591
182. ^
183. ^ Bruhn, Henning; Schaudt, Oliver (2012). "The virtual Haken conjecture". arXiv:1204.2810v1 [math.GT].
184. ^ Lee, Choongbum (2012). "Embedded minimal tori in S^3 and the Lawson conjecture". arXiv:1203.6597v2 [math.DG].
185. ^ Bruhn, Henning; Schaudt, Oliver (2011). "The good pants homology and the Ehrenpreis conjecture". arXiv:1101.1330v4 [math.GT].
186. ^ Bruhn, Henning; Schaudt, Oliver (2009). "Rational group ring elements with kernels having irrational dimension". Proceedings of the London Mathematical Society. 107 (6): 1424–1448. arXiv:0909.2360. Bibcode:2009arXiv0909.2360A. doi:10.1112/plms/pdt029. S2CID 115160094.
187. ^ Lurie, Jacob (2009). "On the classification of topological field theories". Current Developments in Mathematics. 2008: 129–280. arXiv:0905.0465. Bibcode:2009arXiv0905.0465L. doi:10.4310/cdm.2008.v2008.n1.a3. S2CID 115162503.
188. ^ a b "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (PDF) (Press release). Clay Mathematics Institute. March 18, 2010. Archived from the original on March 22, 2010. Retrieved November 13, 2015. The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman.
189. ^ Bruhn, Henning; Schaudt, Oliver (2008). "Completion of the Proof of the Geometrization Conjecture". arXiv:0809.4040 [math.DG].
190. ^ Bruhn, Henning; Schaudt, Oliver (2015). "The Erdos discrepancy problem". arXiv:1509.05363v5 [math.CO].
191. ^ Duncan, John F. R.; Griffin, Michael J.; Ono, Ken (1 December 2015). "Proof of the umbral moonshine conjecture". Research in the Mathematical Sciences. 2 (1): 26. arXiv:1503.01472. Bibcode:2015arXiv150301472D. doi:10.1186/s40687-015-0044-7. S2CID 43589605.
192. ^ Bruhn, Henning; Schaudt, Oliver (2014). "Regularity of Einstein Manifolds and the Codimension 4 Conjecture". arXiv:1406.6534v10 [math.DG].
193. ^ "A Long-Sought Proof, Found and Almost Lost". Quanta Magazine. Natalie Wolchover. March 28, 2017. Archived from the original on April 24, 2017. Retrieved May 2, 2017.
194. ^ Marques, Fernando C.; Neves, André (2013). "Min-max theory and the Willmore conjecture". Annals of Mathematics. 179 (2): 683–782. arXiv:1202.6036. doi:10.4007/annals.2014.179.2.6. S2CID 50742102.
195. ^ Lee, Choongbum (2011). "A counterexample to Beck's conjecture on the discrepancy of three permutations". arXiv:1104.2922 [cs.DM].
196. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-03-27. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
197. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-10-07. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
198. ^ "page 359" (PDF). Archived (PDF) from the original on 2016-03-27. Retrieved 2016-03-18.
199. ^
200. ^ Cilleruelo, Javier (2010). "Generalized Sidon sets". Advances in Mathematics. 225 (5): 2786–2807. doi:10.1016/j.aim.2010.05.010. hdl:10261/31032. S2CID 7385280.
201. ^ Bruhn, Henning; Schaudt, Oliver (2009). "A proof of the Kauffman-Harary Conjecture". Algebr. Geom. Topol. 9 (4): 2027–2039. arXiv:0906.1612. Bibcode:2009arXiv0906.1612M. doi:10.2140/agt.2009.9.2027. S2CID 8447495.
202. ^ Bruhn, Henning; Schaudt, Oliver (2009). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". arXiv:0910.5501v5 [math.GT].
203. ^ Lu, Zhiqin (2007). "Proof of the normal scalar curvature conjecture". arXiv:0711.3510 [math.DG].
204. ^ Dencker, Nils (2006), "The resolution of the Nirenberg–Treves conjecture" (PDF), Annals of Mathematics, 163 (2): 405–444, doi:10.4007/annals.2006.163.405, S2CID 16630732, archived (PDF) from the original on 2018-07-20, retrieved 2019-04-07
205. ^ "Research Awards", Clay Mathematics Institute, archived from the original on 2019-04-07, retrieved 2019-04-07
206. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-04-06. Retrieved 2016-03-22.CS1 maint: archived copy as title (link)
207. ^ "Fields Medal – Ngô Bảo Châu". International Congress of Mathematicians 2010. ICM. 19 August 2010. Archived from the original on 24 September 2015. Retrieved 2015-11-12. Ngô Bảo Châu is being awarded the 2010 Fields Medal for his proof of the Fundamental Lemma in the theory of automorphic forms through the introduction of new algebro-geometric methods.
208. ^ Bruhn, Henning; Schaudt, Oliver (2004). "Tameness of hyperbolic 3-manifolds". arXiv:math/0405568.
209. ^ "Graph Theory". Archived from the original on 2016-03-08. Retrieved 2016-03-18.
210. ^ Chung, Fan; Greene, Curtis; Hutchinson, Joan (April 2015). "Herbert S. Wilf (1931–2012)". Notices of the AMS. 62 (4): 358. doi:10.1090/noti1247. ISSN 1088-9477. OCLC 34550461. The conjecture was finally given an exceptionally elegant proof by A. Marcus and G. Tardos in 2004.
211. ^ "Bombieri and Tao Receive King Faisal Prize" (PDF). Notices of the AMS. 57 (5): 642–643. May 2010. ISSN 1088-9477. OCLC 34550461. Archived (PDF) from the original on 2016-03-04. Retrieved 2016-03-18. Working with Ben Green, he proved there are arbitrarily long arithmetic progressions of prime numbers—a result now known as the Green–Tao theorem.
212. ^ Bruhn, Henning; Schaudt, Oliver (2004). "The classification of Kleinian surface groups, II: The Ending Lamination Conjecture". arXiv:math/0412006.
213. ^ Connelly, Robert; Demaine, Erik D.; Rote, Günter (2003), "Straightening polygonal arcs and convexifying polygonal cycles" (PDF), Discrete and Computational Geometry, 30 (2): 205–239, doi:10.1007/s00454-003-0006-7, MR 1931840, S2CID 40382145
214. ^ Green, Ben (2004), "The Cameron–Erdős conjecture", The Bulletin of the London Mathematical Society, 36 (6): 769–778, arXiv:math.NT/0304058, doi:10.1112/S0024609304003650, MR 2083752, S2CID 119615076
215. ^ "News from 2007". American Mathematical Society. AMS. 31 December 2007. Archived from the original on 17 November 2015. Retrieved 2015-11-13. The 2007 prize also recognizes Green for "his many outstanding results including his resolution of the Cameron-Erdős conjecture..."
216. ^ Voevodsky, Vladimir (2003). "Reduced power operations in motivic cohomology" (PDF). Publications Mathématiques de l'IHÉS. 98: 1–57. arXiv:math/0107109. CiteSeerX 10.1.1.170.4427. doi:10.1007/s10240-003-0009-z. S2CID 8172797. Archived from the original on 2017-07-28. Retrieved 2016-03-18.
217. ^ Savchev, Svetoslav (2005). "Kemnitz' conjecture revisited". Discrete Mathematics. 297 (1–3): 196–201. doi:10.1016/j.disc.2005.02.018.
218. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-03-08. Retrieved 2016-03-23.CS1 maint: archived copy as title (link)
219. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-04-03. Retrieved 2016-03-20.CS1 maint: archived copy as title (link)
220. ^ Chudnovsky, Maria; Robertson, Neil; Seymour, Paul; Thomas, Robin (2002). "The strong perfect graph theorem". arXiv:math/0212070. Unknown parameter |url= ignored (help)
221. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-10-07. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
222. ^ Knight, R. W. (2002), The Vaught Conjecture: A Counterexample, manuscript
223. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-03-03. Retrieved 2016-03-22.CS1 maint: archived copy as title (link)
224. ^ Metsänkylä, Tauno (5 September 2003). "Catalan's conjecture: another old diophantine problem solved" (PDF). Bulletin of the American Mathematical Society. 41 (1): 43–57. doi:10.1090/s0273-0979-03-00993-5. ISSN 0273-0979. Archived (PDF) from the original on 4 March 2016. Retrieved 13 November 2015. The conjecture, which dates back to 1844, was recently proven by the Swiss mathematician Preda Mihăilescu.
225. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-10-07. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
226. ^ "Archived copy" (PDF). Archived from the original (PDF) on 2015-09-08. Retrieved 2016-03-18.CS1 maint: archived copy as title (link)
227. ^ Bruhn, Henning; Schaudt, Oliver (2001). "Deligne's Conjecture on 1-Motives". arXiv:math/0102150.
228. ^ Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001), "On the modularity of elliptic curves over Q: wild 3-adic exercises", Journal of the American Mathematical Society, 14 (4): 843–939, doi:10.1090/S0894-0347-01-00370-8, ISSN 0894-0347, MR 1839918
229. ^ Luca, Florian (2000). "On a conjecture of Erdős and Stewart" (PDF). Mathematics of Computation. 70 (234): 893–897. Bibcode:2001MaCom..70..893L. doi:10.1090/s0025-5718-00-01178-9. Archived (PDF) from the original on 2016-04-02. Retrieved 2016-03-18.
230. ^ "Archived copy" (PDF). Archived (PDF) from the original on 2016-04-02. Retrieved 2016-03-20.CS1 maint: archived copy as title (link)
231. ^ Croot, Ernest S., III (2000), Unit Fractions, Ph.D. thesis, University of Georgia, Athens. Croot, Ernest S., III (2003), "On a coloring conjecture about unit fractions", Annals of Mathematics, 157 (2): 545–556, arXiv:math.NT/0311421, Bibcode:2003math.....11421C, doi:10.4007/annals.2003.157.545, S2CID 13514070
232. ^ Bruhn, Henning; Schaudt, Oliver (1999). "The Honeycomb Conjecture". arXiv:math/9906042.
233. ^ Bruhn, Henning; Schaudt, Oliver (1999). "Proof of the gradient conjecture of R. Thom". arXiv:math/9906212.
234. ^ Ullmo, E (1998). "Positivité et Discrétion des Points Algébriques des Courbes". Annals of Mathematics. 147 (1): 167–179. arXiv:alg-geom/9606017. doi:10.2307/120987. JSTOR 120987. S2CID 119717506. Zbl 0934.14013.
235. ^ Zhang, S.-W. (1998). "Equidistribution of small points on abelian varieties". Annals of Mathematics. 147 (1): 159–165. doi:10.2307/120986. JSTOR 120986.
236. ^ Lafforgue, Laurent (1998), "Chtoucas de Drinfeld et applications" [Drinfelʹd shtukas and applications], Documenta Mathematica (in French), II: 563–570, ISSN 1431-0635, MR 1648105, archived from the original on 2018-04-27, retrieved 2016-03-18
237. ^ Norio Iwase (1 November 1998). "Ganea's Conjecture on Lusternik-Schnirelmann Category". ResearchGate.
238. ^ Merel, Loïc (1996). ""Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields]". Inventiones Mathematicae. 124 (1): 437–449. Bibcode:1996InMat.124..437M. doi:10.1007/s002220050059. MR 1369424. S2CID 3590991.
239. ^ Chen, Zhibo (1996). "Harary's conjectures on integral sum graphs". Discrete Mathematics. 160 (1–3): 241–244. doi:10.1016/0012-365X(95)00163-Q.

### Books discussing unsolved problems

1. ^ The Sverdlovsk Notebook: collects unsolved problems in semigroup theory, Ural State University, 1979
2. ^ The Sverdlovsk Notebook: collects unsolved problems in semigroup theory, Ural State University, 1989
3. ^ Fuks 1974, p. 47, 88, 116, 134, 158, 159, 186, 210, 242, 243, 292, 318.
4. ^ Boltiansky 1965, p. 83.
5. ^ Grunbaum 1971, p. 6.
6. ^ V. G. Vizing Some unresolved problems for Graph theory // Russian Mathematical Surveys, 23:6(144) (1968), 117–134; Russian Math. Surveys, 23:6 (1968), 125–141
7. ^ Sprinjuk 1967, p. 150—154.