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arXiv · 2002.01297

Induced Ramsey number for a star versus a fixed graph

Maria Axenovich·Izolda Gorgol
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Abstract

For graphs G and H, let the induced Ramsey number IR(H,G) be the smallest number of vertices in a graph F such that 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. In this note we consider the case when G=Sn is a star on n edges, for large n, and H is a fixed graph. We prove that (r-1)n < IR(H, Sn) < (r-1)(r-1)n + cn, for any c>0, sufficiently large n, and r 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 by a clique H.

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Maria Axenovich, Izolda Gorgol. 2020-02-04. Induced Ramsey number for a star versus a fixed graph. https://arxiv.org/abs/2002.01297

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