Induced Ramsey numbers for fans

The induced Ramsey number $r_{\mathrm{i}\mathrm{n}\mathrm{d}}(G,H)$ 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 ${\mathrm{r}}_{\text{ind}}(G,F_n)\le C{n}^2$ 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 $K_{1,\mathrm{\ell }}$ $(\mathrm{\ell }\le n)$, constructive coloring and matching arguments yield $2n+2\mathrm{\ell }-1\le {\mathrm{r}}_{\text{ind}}(K_{1,\mathrm{\ell }},F_n)\le (\mathrm{\ell }+n-1)(\mathrm{\ell }+1)+1$, with the exact value ${\mathrm{r}}_{\text{ind}}(K_{1,2},F_n)=3n+4$.