Generalized local colorings of graphs

Let k be a fixed positive integer and let H be a graph with at least k + 1 edges. A local (H, k)-coloring of a graph G is a coloring of the edges of G such that edges of no subgraph of G isomorphic to a subgraph of H are colored with more than k colors. In the paper we investigate properties of local (H, k)-colorings. We prove the Ramsey property for such colorings, establish conditions for the density property and the bipartite version of the Ramsey theorem to hold, and prove the induced variant of the Ramsey theorem with forbidden large cliques.