Mean Ramsey–Turán numbers

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 $RT(n, K_ m, 2 - mean) = RT(n, K_ m, 2)$ 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 $RT(n, H,\rho - mean) \leq RT(n, K_{\chi(H)},\rho - mean) + o(n^2)$ . 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 $RT(n, H, 2 - mean) = RT(n, K_ 3, 2 - mean) + o(n^2) = RT(n, K_ 3, 2) + o(n^2) = 0.4n^2(1 + o(1))$ . © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 126–134, 2006