Asymptotic behavior of generalized Ramsey numbers

Denote by \( F^{(t)}(n, \alpha)\) the largest integer for which it is possible to split the \( t\)-tuples of a set \( S\) of \( n\) elements into \( 2\) classes so that for every \( X \subset S\) with \( \vert X\vert \geq F^{(t)}(n, \alpha)\), each class contains more than \( \alpha \binom{\vert X\vert}{t} \) \( t\)-tuples of \( X\). Note that \( F^{(t)}(n,0)\) is just the usual Ramsey function \( r_t(n,n)\). It is easy to show that for every \( 0 \leq \alpha \leq 1/2\),

\(\displaystyle c(\alpha) \log n < F^{(2)} (n,\alpha) < c'(\alpha) \log n. \)


Prove that \[ F^{(2)}(n,\alpha) \sim c \log n \] for an appropriate \(c\) and determine \(c\).

1 P. Erdös, Problems and results on graphs and hypergraphs: similarities and differences, Mathematics of Ramsey theory, Algorithms Combin., 5, (J. Nešetril and V. Rödl, eds.), 12-28, Springer, Berlin, 1990.