Induced Ramsey Number for a Star Versus a Fixed Graph

We write $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$ for graphs $F, G,$ and $H$, if any coloring of the edges $F$ in red blue, there is either a induced copy $H$ or blue $G$. For $G$ let $\mathrm{IR}(H,G)$ be smallest number vertices graph such that \longrightarrow}(H,G)$.
 In this note we consider case when star on $n$ edges, large fixed graph. prove $$ (\chi(H)-1) n \leq \mathrm{IR}(H, K_{1,n}) (\chi(H)-1)^2n + \epsilon n,$$ $\epsilon>0$, sufficiently $n$, $\chi(H)$ denoting chromatic $H$. The lower bound asymptotically tight bipartite upper attained up to constant factor, example clique.