Induced Ramsey numbers for fans

Abstract

The induced Ramsey number $r_{\mathrm{ind}}(G,H)$ is defined as the minimum order of a graph $F$ on 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 $H$. Motivated by the Kohayakawa-Prömel-Rödl conjecture, we prove that a quadratic upper bound ${\mathrm{r}}_{\text{ind}}\left(G,F_n\right)\le C{n}^2$ for fixed $G$, where $F_n$ is a graph with one central vertex, $2n$ leaf vertices, and $n$ disjoint edges. In particular, for star graphs $K_{1,\ell }$ $(\ell \le n)$, constructive coloring and matching arguments yield $2n+2\ell -1\le {\mathrm{r}}_{\text{ind}}\left(K_{1,\ell },F_n\right)\le (\ell +n-1)(\ell +1)+1$, with the exact value ${\mathrm{r}}_{\text{ind}}\left(K_{1,2},F_n\right)=3n+4$.