The Order of Monochromatic Subgraphs with a Given Minimum Degree

Let G be a graph. For a given positive integer d ,l etfG(d )d enote the largest integer t such that in every coloring of the edges of G with two colors there is a monochromatic subgraph with minimum degree at least d and order at least t .L et fG(d) = 0 in case there is a 2-coloring of the edges of G with no such monochromatic subgraph. Let f(n,k,d) denote the minimum of fG(d )w hereG ranges over all graphs with n vertices and minimum degree at least k. In this paper we establish f(n,k,d) whenever k or n k are fixed, and n is suciently large. We also consider the case where more than two colors are allowed.

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