Open, the site's label; no claim about the problem exists. The only known results are the two-sided bound $c_\alpha'\log n<F(n,\alpha)<c_\alpha''\log n$ for $0\le\alpha<1/2$, asserted without proof by Erdős in 1990 ("The probability method easily gives", display (29)) and by Conlon, Fox and Sudakov in 2008 ("It is easy to show"), and quoted by the site's commentary as $F(n,\alpha)\asymp_\alpha\log n$. No source proving that $F(n,\alpha)/\log n$ converges, or determining $c_\alpha$ for any $\alpha$, was found in the search whose scope the Current assessment records. An observation made here: since $F(n,0)$ is the least $m$ with $R(m,m)>n$, the case $\alpha=0$ of the statement, $F(n,0)\sim c_0\log n$, holds exactly when $\log R(k,k)\sim k/c_0$, that is, when $\lim_{k\to\infty}R(k,k)^{1/k}$ exists (Problem 77), with $c_0$ the reciprocal of the logarithm of that limit; so the problem contains the existence half of Problem 77 as its $\alpha=0$ case. This is a bounded negative finding, not a certificate of openness.