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.

Some of these are the following:

• The Cameron–Erdős conjecture on sum-free sets of integers, proved by Ben Green.
• The Erdős–Burr conjecture on Ramsey numbers of graphs.
• The Erdős–Faber–Lovász conjecture on coloring unions of cliques.
• The Erdős–Graham conjecture in combinatorial number theory on monochromatic Egyptian fraction representations of unity.
• 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. [Ramsey-type theorems, Discrete Applied Mathematics 25 (1989) 37-52]
• The Erdős–Heilbronn conjecture in combinatorial number theory on the number of sums of two sets of residues modulo a prime, proved by J.A. Dias da Silva and Y.O. Hamidoune in 1994.
• The Erdős–Lovász conjecture on weak/strong delta-systems ([1], p. 406), proved by Michel Deza.
• The Erdős–Mollin–Walsh conjecture on consecutive triples of powerful numbers.
• The Erdős–Menger conjecture on disjoint paths in infinite graphs. (solved by Ron Aharoni and Eli Berger])
• The Erdős–Selfridge conjecture that a covering set contains at least one odd member.
• The Erdős–Stewart conjecture on the Diophantine equation n! + 1 = p_k^a p_k+1^b (solved by Luca, MR2001g:11042)
• 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–Woods conjecture on numbers determined by the set of prime divisors of the following k numbers.
• 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 on equitable colorings proven in 1970 by András Hajnal and Endre Szemerédi and now known as the Hajnal–Szemerédi theorem.
• A conjecture with Norman Oler on circle packing in an equilateral triangle with a number of circles one less than a triangular number.

See also

• List of topics named after Paul Erdős

External links

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