I have a question that I need help with. Suppose you have some integers $k,q$ and $N$ and a fixed q-coloring on the edges of $K_N$ such that it contains no monochromatic $K_k$. Is there a nice way to bound the number of possible monochromatic $K_i$ for $i<k$ and a fixed color? So for $i=2$ we know every two vertices form a monochromatic K_2 so that's pretty useless. But what about $K_{k-1}$ for example? Intuitively it seems like there can't be a lot of them in the same color. I tried to find a bound using an induction argument but I'm stuck & a quick internet search didn't help. Maybe someone knows an answer to this?
2026-03-26 21:27:14.1774560434
Bound on the number of monochromatic $K_i$ in a $K_k$ free colored graph
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