Ramsey numbers of cycles versus general graphs

Abstract

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 $n\geq \lvert H\rvert \chi (H)$ . We improve this 40-year-old result of Burr by giving quantitative bounds of the form $n\geq C\lvert H\rvert \log ^4\chi (H)$ , 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.