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

Fix a graph F and a positive integer r. With a graph G, we associate the quantity cr,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 cr,F:N⟶N given by cr,F(n)=max{cr,F(G):|V(G)|=n}, the maximum of cr,F(G) over all graphs G on n vertices. In this paper we study the asymptotic behavior of cr,F(n) and describe the extremal graphs for some forbidden graphs F with linear Turán number, such as small paths, stars and trees.