A Solution to the Kohayakawa-Prömel-Rödl Conjecture

The induced Ramsey number $r_{\mathrm{i}\mathrm{n}\mathrm{d}}(G,H)$ 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 $r_{\mathrm{i}\mathrm{n}\mathrm{d}}(G,H)$ is bounded by $\mathrm{\mid }V(H){\mathrm{\mid }}^f$ 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 $\mathrm{\mid }V(G)\mathrm{\mid }=k\ge 2$ and $\mathrm{\mid }V(H)\mathrm{\mid }=t\ge 2$, $r_{\mathrm{i}\mathrm{n}\mathrm{d}}(G,H)\le t^{60k^2+3}$. This verifies the conjecture with $f(G)=60\mathrm{\mid }V(G){\mathrm{\mid }}^2+3$.