Ramsey numbers of cycles versus general graphs

TL;DR

This paper improves bounds on Ramsey numbers involving cycles and general graphs, confirming a conjecture for large chromatic number graphs by providing near-optimal quantitative bounds.

Contribution

It refines Burr's 40-year-old result by establishing near-optimal bounds for the Ramsey number involving cycles and arbitrary graphs, confirming a conjecture for large chromatic number cases.

Findings

Established bounds of the form n ≥ C|H| log^4 χ(H)

Proved the conjecture for all graphs with large chromatic number

Improved previous results by quantifying the bounds

Abstract

The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$, provided $n$ is sufficiently large, a natural lower bound construction gives the correct Ramsey number involving cycles: $R(C_n,H)=(n-1)(\chi(H)-1)+\sigma(H)$, where $\sigma(H)$ is the minimum possible size of a colour class in a $\chi(H)$-colouring of $H$. Allen, Brightwell and Skokan conjectured that the same should be true already when $n\geq |H|\chi(H)$. We improve this 40-year-old result of Burr by giving quantitative bounds of the form $n\geq C|H|\log^4\chi(H)$, which is optimal up to the logarithmic factor. In particular, this proves a strengthening of the Allen-Brightwell-Skokan conjecture for all graphs $H$ with large chromatic number.