Erdős Problem 563

Let $F(n,\alpha)$ denote the smallest $m$ such that there exists a $2$-colouring of the edges of $K_n$ so that every $X\subseteq [n]$ with $\lvert X\rvert\geq m$ contains more than $\alpha \binom{\lvert X\rvert}{2}$ many edges of each colour.

Prove that, for every $0\leq \alpha < 1/2$, $$F(n,\alpha)\sim c_\alpha\log n$$ for some constant $c_\alpha$ depending only on $\alpha$.

This problem is #39 in Ramsey Theory in the graphs problem collection.