Edge-colorings of graphs avoiding fixed monochromatic subgraphs with linear Turلn number

Fix a graph F and a positive integer r . With a graph G , we associate the quantity c r , F ( G ) , the number of r -colorings of the edge set of G with no monochromatic copy of F as a subgraph. Consider the function c r , F : N ⟶ N given by c r , F ( n ) = max { c r , F ( G ) : | V ( G ) | = n } , the maximum of c r , F ( G ) over all graphs G on n vertices. In this paper we study the asymptotic behavior of c r , F ( n ) and describe the extremal graphs for some forbidden graphs F with linear Turán number, such as small paths, stars and trees.