| Abstract: | A ρ‐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 ρ. For a graph H and for ρ ≥ 1, the mean Ramsey–Turán number RT(n, H,ρ ? mean) is the maximum number of edges a ρ‐mean colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. It is conjectured that  where  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  . We also prove that  . 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  . © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 126–134, 2006 |
