List of conjectures by Paul Erdős

Template:Short description 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 $2^{n}$ contains at least one digit 2 for every $n > 8$ .^[3]
The conjecture that the Erdős–Moser equation, Template:Math, has no solutions except Template:Math.

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 $\sum_{n \in A} \frac{1}{n \log n}$ 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]

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List of conjectures by Paul Erdős

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List of conjectures by Paul Erdős

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References

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