Prove that for every fixed $0\leq \alpha \leq 1/2$, as $n\to\infty$, \[F(n,\alpha)\sim c_\alpha \log n\] for some constant $c_\alpha$.

OPEN

Let $\alpha>0$ and $n\geq 1$. Let $F(n,\alpha)$ be the largest $k$ such that in any 2-colouring of the edges of $K_n$ any subgraph $H$ on at least $k$ vertices contains more than $\alpha\binom{\lvert H\rvert}{2}$ many edges of each colour.

Prove that for every fixed $0\leq \alpha \leq 1/2$, as $n\to\infty$, \[F(n,\alpha)\sim c_\alpha \log n\] for some constant $c_\alpha$.

It is easy to show with the probabilistic method that there exist $c_1(\alpha),c_2(\alpha)$ such that
\[c_1(\alpha)\log n < F(n,\alpha) < c_2(\alpha)\log n.\]