Abstract
We write $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$ for graphs $F, G,$ and $H$, if for any coloring of the edges of $F$ in red and blue, there is either a red induced copy of $H$ or a blue induced copy of $G$. For graphs $G$ and $H$, let $\mathrm{IR}(H,G)$ be the smallest number of vertices in a graph $F$ such that $F{\buildrel {\text{ind}} \over \longrightarrow}(H,G)$.
 In this note we consider the case when $G$ is a star on $n$ edges, for large $n$ and $H$ is a fixed graph. We prove that $$ (\chi(H)-1) n \leq \mathrm{IR}(H, K_{1,n}) \leq (\chi(H)-1)^2n + \epsilon n,$$ for any $\epsilon>0$, sufficiently large $n$, and $\chi(H)$ denoting the chromatic number of $H$. The lower bound is asymptotically tight for any fixed bipartite $H$. The upper bound is attained up to a constant factor, for example when $H$ is a clique.
Highlights
We write F −in→d (H, G) for graphs F, G, and H, if for any coloring of the edges of F in red and blue, there is either a red induced copy of H or a blue induced copy of G
When H is a blow-up of H, we proceed by taking G to be a blow-up of H with parts of sizes s, s((n − 1)d2 + 1), . . . , s((n − 1)di + 1), . . . and embedding a part of H corresponding to the blow-up of vi in Vi
Observe first that G contains an independent set of size n, call it A, otherwise there is no induced K1,n in G and all edges could be colored blue
Summary
We write F −in→d (H, G) for graphs F, G, and H, if for any coloring of the edges of F in red and blue, there is either a red induced copy of H or a blue induced copy of G. For graphs G and H, the induced Ramsey number for H and G, denoted by IR(H, G), is the smallest number of vertices in a graph F such that F −in→d (H, G) The existence of such a graph F for any graphs H and G was first proven by Deuber [4], extending a classical result by Ramsey [17]. In this note we consider the case when G is a star on n edges, i.e., G = K1,n, and H is a fixed graph. The following gives more precise upper bounds for some bipartite graphs that allows to replace the γn term of Theorem 2 with a o(n) term. For standard graph theoretic notions and terminology, we refer the reader to West [19]
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