Mean Ramsey-Tur\'an 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 $\rho \geq 1$, the {\em mean Ramsey-Tur\'an 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 $RT(n,H,\rho-mean) \leq RT(n,K_{\chi(H)},\rho-mean)+o(n^2)$. 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 $RT(n,H,2-mean) = RT(n,K_3,2-mean)+o(n^2)=RT(n,K_3,2)+o(n^2)=0.4n^2(1+o(1))$.