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 ${\displaystyle 2^{n}}$  contains at least one digit 2 for every ${\displaystyle n>8}$ .^[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 = p_k^a p_k+1^b, 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 ${\displaystyle \sum _{n\in A}{\frac {1}{n\log n}}}$  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]

See also

• List of things named after Paul Erdős

References

1. 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.
2. Oler, Norman (1961), "A finite packing problem", Canadian Mathematical Bulletin, 4 (2): 153–155, doi:10.4153/CMB-1961-018-7, MR 0133065.
3. 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
4. Houston-Edwards, Kelsey (5 April 2021), "Mathematicians Settle Erdős Coloring Conjecture", Quanta Magazine, retrieved 2021-04-05
5. 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.
6. Kalai, Gil (May 22, 2015), "Choongbum Lee proved the Burr-Erdős conjecture", Combinatorics and more, retrieved 2015-05-22
7. 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
8. 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.
9. 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.
10. 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.
11. 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.
12. 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.
13. 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.
14. 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.
15. 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.
16. 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.
17. Ford, Kevin; Green, Ben; Konyagin, Sergei; Tao, Terence (2016), "Large gaps between consecutive prime numbers", Annals of Mathematics, Second series, 183 (3): 935–974, arXiv:1408.4505, doi:10.4007/annals.2016.183.3.4
18. 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.
19. 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
20. 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
21. Lichtman, Jared Duker (2022-02-04). "A proof of the Erdős primitive set conjecture". arXiv:2202.02384 [math.NT].
22. Cepelewicz, Jordana (2022-06-06). "Graduate Student's Side Project Proves Prime Number Conjecture". Quanta Magazine. Retrieved 2022-06-06.
23. Haran, Brady. "Primes and Primitive Sets". Numberphile. Retrieved 2022-06-21.
24. Janzer, Oliver; Sudakov, Benny (2022-04-26). "Resolution of the Erdős-Sauer problem on regular subgraphs". arXiv:2204.12455 [math.CO].
25. "New Proof Shows When Structure Must Emerge in Graphs". Quanta Magazine. 2022-06-23. Retrieved 2022-06-26.

External links

• Fan Chung, "Open problems of Paul Erdős in graph theory"
• Fan Chung, living version of "Open problems of Paul Erdős in graph theory"