Local and Mean Ramsey Numbers for Trees

The usual Ramsey number R(G, k) is the smallest positive integer n such that any coloring of the edges of Kn by at most k colors contains a monochromatic copy of G. Over the past years several papers have been written in which the number of different colors used has no longer been restricted to k, but a restriction is placed on the number of colored edges incident to the vertices. To be precise, let H be a fixed graph of order n and let f be a coloring on the edges of H. For each v # V(H) define kf (v) as the number of distinct colors that appear on the edges of H incident to v. The coloring f is called a local k-coloring if kf (v) k for all v # V(H), and is called a mean k-coloring if (1 n) v kf (v) k. Further, the local k-Ramsey number R(G, k loc) is defined as the smallest positive integer n such that doi:10.1006 jctb.2000.1950, available online at http: www.idealibrary.com on