Ramsey numbers of cycles versus general graphs

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)( x(H) - 1) + sigma (H), where sigma(H) is the minimum possible size of a colour class in a x(H)-colouring of H. Allen, Brightwell and Skokan conjectured that the same should be true already when n >= |H|x(H).We improve this 40-year-old result of Burr by giving quantitative bounds of the form n >= C|H| log(4 )x (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.