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. 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 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 . 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. 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. The Burr–Erdős conjecture on Ramsey numbers of graphs, proved by Choongbum Lee in 2015. A conjecture on equitable colorings proven in 1970 by András Hajnal and Endre Szemerédi and now known as the Hajnal–Szemerédi theorem. 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. The Erdős–Lovász conjecture on weak/strong delta-systems, proved by Michel Deza in 1974. 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. The Erdős–Graham conjecture in combinatorial number theory