List of conjectures by Paul Erdős
From Infogalactic: the planetary knowledge core
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 = pka pk+1b, 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
References
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