Effective Bounds for Induced Size-Ramsey Numbers of Cycles

The induced size-Ramsey number \(\hat{r}_\text {ind}^k(H)\) 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., \(\hat{r}_\text {ind}^k(C_n)\le Cn\) 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 \(\hat{r}_\text {ind}^k(C_n)\le O(k^{102})n\) when n is even, thus obtaining only a polynomial dependence of C on k. We also prove \(\hat{r}_\text {ind}^k(C_n)\le e^{O(k\log k)}n\) for odd n, which almost matches the lower bound of \(e^{\Omega (k)}n\). Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies \(\hat{r}^k(C_n)=e^{O(k)}n\) 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.