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 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 on whether the ternary expansion of ${\displaystyle 2^{n}}$ contains at least one digit 2, for ${\displaystyle n>8}$.^[2]

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.^[3]
• 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.^[4]
• 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.^[5]
• 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.^[6]
• The on weak/strong delta-systems, proved by Michel Deza in 1974.^[7]
• 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.^[8]
• The Erdős–Graham conjecture in combinatorial number theory on monochromatic Egyptian fraction representations of unity, proved by Ernie Croot in 2000.^[9]
• The on the Diophantine equation n! + 1 = p_k^a p_k+1^b, solved by Florian Luca in 2001.^[10]
• The Cameron–Erdős conjecture on sum-free sets of integers, proved by Ben Green and Alexander Sapozhenko in 2003–2004.^[11]
• The Erdős–Menger conjecture on disjoint paths in infinite graphs, proved by Ron Aharoni and Eli Berger in 2009.^[12]
• 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.^[13]
• The Erdős–Rankin conjecture on prime gaps, proved by Ford, Green, Konyagin, and Tao in 2014.
• Erdős discrepancy problem on partial sums of ±1-sequences.
• Erdős squarefree conjecture that central binomial coefficients C(2n, n) are never squarefree for n > 4 was proved in 1996.

See also[]

• List of things named after Paul Erdős

References[]

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"