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 p > 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, Km, 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 K₃. We also prove that RT(n, H,ρ- mean) > RT(n, K_χ(H), p - mean) + o(n²). 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, K₃, 2 - mean) + o(n²) = RT(n. K³. 2) + o(n²) = 0.4n²(1 + o(1)).