Given a graph on $n$ vertices find the maximum amount of edges so it can be colored with no monochromatic $K_m$ I invented a problem and I wanted to share :What is the maximum amount of edges a graph on $n$ vertices can have if it can be edge-colored with $k$ colors so that it does not have a monochromatic $K_m$? 
I think I have a solution, I would like a verification. Generalizations would also be greatly appreciated.
Thank you very much in advance.
Regards.
 A: Such a graph cannot contain a complete subgraph of size $R\underbrace{(m,m\dots m)}_{\text{k times}}=j$ . What is the subgraph on $n$ vertices that has the most edges and does not contain a complete sugraph of size $n$? It is the Turan graph $T(n,j-1)$, so if $T(n,j-1)$ works we are done.
By the definition of the Ramsey number $K_{j-1}$ admits a coloring $C$ with $k$ colors and no monochromatic $K_m$. So what we do is take $T(n,j-1)$ and assume each of the $j-1$ parts of $T(n,j-1)$ is a vertex in $K_{j-1}$. So given two adjacent vertices of $T(n,j-1)$ we color that edge using the color used in $C$ to connect the vertices of their corresponding parts (Since every part of $T(n,j-1)$ is seen as a vertex of $K_{j-1}$. This coloring gives us no monochromatic $K_m$ because if we have $m$ vertices which are all pairwise connected they must all be in different parts, and then the colors of the edges between them will be the same as the color of the edges between the vertices in $K_{j-1}$ with the coloring $C$.
Hence the maximum number of edges is the number of edges in the $T(n,j-1)$ which is $\lfloor\frac{(j-2)n^2}{2(j-1)}\rfloor$. It also follows from Turan the graph is unique.
