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

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