A Solution to the Kohayakawa-Prömel-Rödl Conjecture

Abstract

The induced Ramsey number $r_{\mathrm{ind}}(G,H)$ is defined as the minimum order of a graph $F$ such that any 2-coloring of its edges with red and blue leads to either a red induced copy of $G$ or a blue induced copy of $H$. In 1998, Kohayakawa, Prömel, and Rödl conjectured that for any fixed graph $G$, there exists a constant $f=f(G)$ such that the induced Ramsey number $r_{\mathrm{ind}}(G,H)$ is bounded by $|V(H){|}^f$ for every graph $H$. In this paper, we completely solve the Kohayakawa--Prömel--Rödl conjecture, which states that for any graphs $G$ and $H$ with $|V(G)|=k\ge 2$ and $|V(H)|=t\ge 2$, $r_{\mathrm{ind}}(G,H)\le t^{60k^2+3}$. This verifies the conjecture with $f(G)=60|V(G){|}^2+3$.