Ramsey numbers of cycles versus general graphs

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

Towards the Erdős-Gallai cycle decomposition conjecture

“…Sublinear expansion is a weaker notion of this classical expansion introduced by Komlós and Szemerédi [34,35], where we take a much smaller value of λ, but which is significant as every graph contains a sublinear expander H with λ = Θ(1/ log 2 |H|) (and even has a nice decomposition into sublinear expanders, as we will prove and use). Komlós and Szemerédi used sublinear expansion to find minors in sparse graphs, and more recently sublinear expansion has found a host of other applications (see, for example, [8,21,22,26,27,33,39,40,41,42,45,50]).…”

How to build a pillar: a proof of Thomassen's conjecture

“…There has been a sequence of advancements on the theory of sublinear expanders, which results in resolutions of several long-standing conjectures. We refer the interested readers to [10,11,12,13,14,19,20,21].…”

Disjoint isomorphic balanced clique subdivisions

“…Our approach uses a version of expander called sublinear expanders. For more recent applications of the theory of sublinear expanders, we refer the interested readers to [6,5,9,7,8,10,13,14,15].…”