Ramsey numbers for local colorings

The concept of a local k-coloring of a graph G is introduced and the corresponding local k-Ramsey number r 1 ~c(G) is considered. A local k-coloring of G is a coloring ofits edges in such a way that the edges incident to any vertex of G are colored with at most k colors. The number r 1 ~c(G) is the minimum m for which Km contains a monochromatic copy of G for every local k-coloring of Km. The number r 1 ~c(G) is a natural generalization of the usual Ramsey number rk(G) defined for usual k-colorings. The results reflect the relationship between rk(G) and r 1 ~c(G) for certain classes of graphs.