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Erdős #1030 · graph theory / ramsey theory

Let R(k,l)R(k,l) be the usual Ramsey number: the smallest nn such that if the edges of KnK_n are coloured red and blue then there exists either a red KkK_k or a blue KlK_l.

Prove the existence of some c>0c>0 such that

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MSC 05C · Graph theory

Inferred from the community field; finer classification has not been reviewed.

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Problem statement

Let R(k,l)R(k,l) be the usual Ramsey number: the smallest nn such that if the edges of KnK_n are coloured red and blue then there exists either a red KkK_k or a blue KlK_l.

Prove the existence of some c>0c>0 such that

lim⁡k→∞R(k+1,k)R(k,k)>1+c.\lim_{k\to \infty}\frac{R(k+1,k)}{R(k,k)}> 1+c.

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Background

A problem of Erdős and Sós, who could not even prove whether R(k+1,k)−R(k,k)>kcR(k+1,k)-R(k,k)>k^c for any c>1c>1.

It is trivial that R(k+1,k)−R(k,k)≥k−2R(k+1,k)-R(k,k)\geq k-2. Burr, Erdős, Faudree, and Schelp [BEFS89] proved

R(k+1,k)−R(k,k)≥2k−5.R(k+1,k)-R(k,k)\geq 2k-5.

See also [544] for a similar question concerning R(3,k)R(3,k), and [1014] for the general off-diagonal case.

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References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1030 link
  2. [2]Formal Conjectures — Erdős Problem 1030 link

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