The Order of Monochromatic Subgraphs with a Given Minimum Degree - Caro, Yair; Yuster, Raphael (The Electronic Journal of Combinatorics, 2003)
Let G be a graph. For a given positive integer d, let f_G(d) denote 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. Let f_G(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 f_G(d) where G 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 sufficiently large. We also consider the case where more than two colors are allowed.