Mean Ramsey-Turan numbers

A rho-mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most rho. For a graph H and for rho >= 1, the mean Ramsey-Turan number RT(n, H, rho - mean) is the maximum number of edges a rho-mean colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. It is conjectured that RT(n, K-m, 2 - mean) = RT(n, K-m, 2) where RT(n, H, k) is the maximum number of edges a k edge-colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. We prove the conjecture holds for K-3. We also prove that RT(n, H, rho - mean) <= RT(n, K-chi(H), rho - mean) + omicron(n(2)). This result is tight for graphs H whose clique number equals their chromatic number. In particular, we get that if H is a 3-chromatic graph having a triangle then RT(n, H, 2 - mean) = RT(n, K-3, 2 - mean) + omicron(n2) = RT(n, K-3, 2) + omicron(n(2)) = 0.4n(2)(1 + omicron(1)). (C) 2006 Wiley Periodicals, Inc.