List of conjectures by Paul Erdős
The prolific mathematician Paul Erdős and his various collaborators made many famous mathematical conjectures, over a wide field of subjects, and in many cases Erdős offered monetary rewards for solving them.
Unsolved
- The Erdős–Gyárfás conjecture on cycles with lengths equal to a power of two in graphs with minimum degree 3.
- The Erdős–Hajnal conjecture that in a family of graphs defined by an excluded induced subgraph, every graph has either a large clique or a large independent set.[1]
- The Erdős–Mollin–Walsh conjecture on consecutive triples of powerful numbers.
- The Erdős–Selfridge conjecture that a covering system with distinct moduli contains at least one even modulus.
- The Erdős–Straus conjecture on the Diophantine equation 4/n = 1/x + 1/y + 1/z.
- The Erdős conjecture on arithmetic progressions in sequences with divergent sums of reciprocals.
- The Erdős–Szekeres conjecture on the number of points needed to ensure that a point set contains a large convex polygon.
- The Erdős–Turán conjecture on additive bases of natural numbers.
- A conjecture on quickly growing integer sequences with rational reciprocal series.
- A conjecture with Norman Oler[2] on circle packing in an equilateral triangle with a number of circles one less than a triangular number.
- The minimum overlap problem to estimate the limit of M(n).
- A conjecture that the ternary expansion of contains at least one digit 2 for every .[3]
Solved
- The Erdős–Faber–Lovász conjecture on coloring unions of cliques, proved (for all large n) by Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus.[4]
- The Erdős sumset conjecture on sets, proven by Joel Moreira, Florian Karl Richter, Donald Robertson in 2018. The proof has appeared in "Annals of Mathematics" in March 2019.[5]
- The Burr–Erdős conjecture on Ramsey numbers of graphs, proved by Choongbum Lee in 2015.[6][7]
- A conjecture on equitable colorings proven in 1970 by András Hajnal and Endre Szemerédi and now known as the Hajnal–Szemerédi theorem.[8]
- A conjecture that would have strengthened the Furstenberg–Sárközy theorem to state that the number of elements in a square-difference-free set of positive integers could only exceed the square root of its largest value by a polylogarithmic factor, disproved by András Sárközy in 1978.[9]
- The Erdős–Lovász conjecture on weak/strong delta-systems, proved by Michel Deza in 1974.[10]
- The Erdős–Heilbronn conjecture in combinatorial number theory on the number of sums of two sets of residues modulo a prime, proved by Dias da Silva and Hamidoune in 1994.[11]
- The Erdős–Graham conjecture in combinatorial number theory on monochromatic Egyptian fraction representations of unity, proved by Ernie Croot in 2000.[12]
- The Erdős–Stewart conjecture on the Diophantine equation n! + 1 = pka pk+1b, solved by Florian Luca in 2001.[13]
- The Cameron–Erdős conjecture on sum-free sets of integers, proved by Ben Green and Alexander Sapozhenko in 2003–2004.[14]
- The Erdős–Menger conjecture on disjoint paths in infinite graphs, proved by Ron Aharoni and Eli Berger in 2009.[15]
- The Erdős distinct distances problem. The correct exponent was proved in 2010 by Larry Guth and Nets Katz, but the correct power of log n is still undetermined.[16]
- The Erdős–Rankin conjecture on prime gaps, proved by Ford, Green, Konyagin, and Tao in 2014.[17]
- The Erdős discrepancy problem on partial sums of ±1-sequences. Terence Tao announced a solution in September 2015; it was published in 2016.[18]
- The Erdős squarefree conjecture that central binomial coefficients C(2n, n) are never squarefree for n > 4 was proved in 1996.[19][20]
- The Erdős primitive set conjecture that the sum for any primitive set A (a set where no member of the set divides another member) attains its maximum at the set of primes numbers, proved by Jared Duker Lichtman in 2022.[21][22][23]
- The Erdős-Sauer problem about maximum number of edges an n-vertex graph can have without containing a k-regular subgraph, solved by Oliver Janzer and Benny Sudakov[24][25]
References
- Erdős, P.; Hajnal, A. (1989), "Ramsey-type theorems", Combinatorics and complexity (Chicago, IL, 1987), Discrete Applied Mathematics, 25 (1–2): 37–52, doi:10.1016/0166-218X(89)90045-0, MR 1031262.
- Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin, 4 (2): 153–155, doi:10.4153/CMB-1961-018-7, MR 0133065.
- Lagarias, Jeffrey C. (2009), "Ternary expansions of powers of 2", Journal of the London Mathematical Society, Second Series, 79 (3): 562–588, arXiv:math/0512006, doi:10.1112/jlms/jdn080, MR 2506687, S2CID 15615918
- Houston-Edwards, Kelsey (5 April 2021), "Mathematicians Settle Erdős Coloring Conjecture", Quanta Magazine, retrieved 2021-04-05
- Moreira, J.; Richter, F. K.; Robertson, D. (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, MR 3919363, S2CID 119158401, Zbl 1407.05236.
- Kalai, Gil (May 22, 2015), "Choongbum Lee proved the Burr-Erdős conjecture", Combinatorics and more, retrieved 2015-05-22
- 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
- Hajnal, A.; Szemerédi, E. (1970), "Proof of a conjecture of P. Erdős", Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969), North-Holland, pp. 601–623, MR 0297607.
- Sárközy, A. (1978), "On difference sets of sequences of integers. II", Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae, 21: 45–53 (1979), MR 0536201.
- Deza, M. (1974), "Solution d'un problème de Erdős-Lovász", Journal of Combinatorial Theory, Series B (in French), 16 (2): 166–167, doi:10.1016/0095-8956(74)90059-8, MR 0337635.
- da Silva, Dias; A., J.; Hamidoune, Y. O. (1994), "Cyclic spaces for Grassmann derivatives and additive theory", Bulletin of the London Mathematical Society, 26 (2): 140–146, doi:10.1112/blms/26.2.140.
- Croot, Ernest S., III (2000), Unit Fractions, Ph.D. thesis, University of Georgia, Athens
{{citation}}: CS1 maint: multiple names: authors list (link). 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{{citation}}: CS1 maint: multiple names: authors list (link). - Luca, Florian (2001), "On a conjecture of Erdős and Stewart", Mathematics of Computation, 70 (234): 893–896, Bibcode:2001MaCom..70..893L, doi:10.1090/S0025-5718-00-01178-9, MR 1677411.
- Sapozhenko, A. A. (2003), "The Cameron-Erdős conjecture", Doklady Akademii Nauk, 393 (6): 749–752, MR 2088503. Green, Ben (2004), "The Cameron-Erdős conjecture", Bulletin of the London Mathematical Society, 36 (6): 769–778, arXiv:math.NT/0304058, doi:10.1112/S0024609304003650, MR 2083752, S2CID 119615076.
- Aharoni, Ron; Berger, Eli (2009), "Menger's Theorem for infinite graphs", Inventiones Mathematicae, 176 (1): 1–62, arXiv:math/0509397, Bibcode:2009InMat.176....1A, doi:10.1007/s00222-008-0157-3, S2CID 15355399.
- Guth, Larry; Katz, Nets H. (2015), "On the Erdős distinct distances problem in the plane", Annals of Mathematics, Second series, 181 (1): 155–190, arXiv:1011.4105, doi:10.4007/annals.2015.181.1.2.
- Ford, Kevin; Green, Ben; Konyagin, Sergei; Tao, Terence (2016), "Large gaps between consecutive prime numbers", Annals of Mathematics, Second series, 183 (3): 935–974, doi:10.4007/annals.2016.183.3.4
- Tao, Terence (2016). "The Erdős discrepancy problem". Discrete Analysis: 1–29. arXiv:1509.05363. doi:10.19086/da.609. ISSN 2397-3129. MR 3533300. S2CID 59361755.
- Sárközy, A. (1985), "On divisors of binomial coefficients. I", Journal of Number Theory, 20 (1): 70–80, doi:10.1016/0022-314X(85)90017-4, MR 0777971
- Ramaré, Olivier; Granville, Andrew (1996), "Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients", Mathematika, 43 (1): 73–107, doi:10.1112/S0025579300011608
- Lichtman, Jared Duker (2022-02-04). "A proof of the Erdős primitive set conjecture". arXiv:2202.02384 [math.NT].
- Cepelewicz, Jordana (2022-06-06). "Graduate Student's Side Project Proves Prime Number Conjecture". Quanta Magazine. Retrieved 2022-06-06.
- Haran, Brady. "Primes and Primitive Sets". Numberphile. Retrieved 2022-06-21.
- Janzer, Oliver; Sudakov, Benny (2022-04-26). "Resolution of the Erdős-Sauer problem on regular subgraphs". arXiv:2204.12455 [math.CO].
- "New Proof Shows When Structure Must Emerge in Graphs". Quanta Magazine. 2022-06-23. Retrieved 2022-06-26.
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