Erdős Problem #563
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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A 2-coloring of \(K_n\) (represented by graph \(G\) on \(\mathrm{Fin}\; n\)) is balanced on subsets of size at least \(m\) with parameter \(\alpha\) if every \(X \subseteq [n]\) with \(|X| \geq m\) contains more than \(\alpha \binom{|X|}{2}\) edges of each color.
\(F(n,\alpha)\) is the smallest \(m\) such that there exists a 2-coloring of the edges of \(K_n\) so that every \(X\subseteq [n]\) with \(|X|\geq m\) contains more than \(\alpha\binom{|X|}{2}\) edges of each color.
Attempts
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Proved the exact piecewise formula R_alpha(4)=18,10,6, computed the complete n<=7 extremal table, and reduced the asymptotic question to existence of lim_k log R_alpha(k)/k, whose alpha=0 case is the
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