The order of monochromatic subgraphs with a given minimum degree

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.