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–Burr conjecture on Ramsey numbers of graphs.
• The Erdős–Faber–Lovász conjecture on coloring unions of cliques.
• 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 set contains at least one odd member.
• 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 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).
• Erdős discrepancy problem on partial sums of ±1-sequences.
  • In September 2015, Terence Tao submitted a proof of this conjecture, which is currently under review

Solved

• A conjecture on equitable colorings proven in 1970 by András Hajnal and Endre Szemerédi and now known as the Hajnal–Szemerédi theorem.^[2]
• The Erdős–Lovász conjecture on weak/strong delta-systems, proved by Michel Deza in 1974.^[3]
• 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.^[4]
• The Erdős–Graham conjecture in combinatorial number theory on monochromatic Egyptian fraction representations of unity, proved by Ernie Croot in 2000.^[5]
• The Erdős–Stewart conjecture on the Diophantine equation n! + 1 = p_k^a p_k+1^b, solved by Florian Luca in 2001.^[6]
• The Cameron–Erdős conjecture on sum-free sets of integers, proved by Ben Green and Alexander Sapozhenko in 2003–2004.^[7]
• The Erdős–Menger conjecture on disjoint paths in infinite graphs, proved by Ron Aharoni and Eli Berger in 2009.^[8]
• 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 open.^[9]

See also

• List of topics named after Paul Erdős

References

1. Erdős, P.; Hajnal, A. (1989), "Ramsey-type theorems", Combinatorics and complexity (Chicago, IL, 1987), Discrete Appl. Math. 25 (1-2): 37–52, doi:10.1016/0166-218X(89)90045-0, MR 1031262.
2. 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.
3. Deza, M. (1974), "Solution d'un problème de Erdős-Lovász", Journal of Combinatorial Theory, Series B (in French) 16: 166–167, doi:10.1016/0095-8956(74)90059-8, MR 0337635.
4. 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.
5. 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, doi:10.4007/annals.2003.157.545.
6. Luca, Florian (2001), "On a conjecture of Erdős and Stewart", Mathematics of Computation 70 (234): 893–896, doi:10.1090/S0025-5718-00-01178-9, MR 1677411.
7. 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.
8. Aharoni, Ron; Berger, Eli (2009), "Menger's Theorem for infinite graphs", Inventiones Mathematicae 176: 1–62, doi:10.1007/s00222-008-0157-3.
9. Guth, l.; Katz, N. H. (2010), On the Erdős distinct distance problem on the plane, arXiv:1011.4105.

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"