Edge colorings of graphs avoiding some fixed monochromatic subgraph with linear Turán number

Let F be a graph and k be a positive integer. With a graph G, we associate the quantity ck,F(G), the number of k-colorings of the edge set of G with no monochromatic copy of F. Consider the function ck,F:N→N given by ck,F(n)=max{ck,F(G):|V(G)|=n}, the maximum of ck,F(G) over all graphs G on n vertices. In this paper we study the asymptotic behavior of ck,F and describe the extremal graphs for the case in which F is a matching, a short path or a star.