Effective bounds for induced size-Ramsey numbers of cycles

Abstract

The induced size-Ramsey number rk ind (H) of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., rk ind (C n ) ≤ Cn for some C = C(k). The constant C comes from the use of the regularity lemma, and has a tower type dependence on k. In this paper we significantly improve these bounds, showing that rk)n when n is even, thus obtaining only a polynomial dependence of C on k. We also prove rk ind (C n ) ≤ e O(k log k) n for odd n, which almost matches the lower bound of e Ω(k) n. Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies rk (C n ) = e O(k) n for odd n. This substantially improves the best previous result of e O(k 2 ) n, and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity.

Short proofs for long induced paths

“…Here we consider a problem on induced size-Ramsey numbers, combining two well-studied extensions of the classical graph Ramsey problem, in which one colours some 'host' graph G and seeks a monochromatic copy of some 'target' graph H. In the induced Ramsey problem, one seeks monochromatic induced copies of H and in the size-Ramsey problem one aims to minimise the size e(G) of G. Induced size-Ramsey problems combine both of these features: the k-colour induced size-Ramsey number rk ind (H) is the smallest integer m such that there exists a graph G on m edges such that every k-colouring of the edges of G contains a monochromatic copy of H. The main open problem in this direction is an old conjecture of Erdős that for any graph H on n vertices one has r2 ind (H) ≤ 2 O(n) . For more background we refer the reader to the survey [7] and recent papers [5,14].…”