Erdős Problem #161

Let \(\alpha\in[0,1/2)\) and \(n,t\geq 1\). Let \(F^{(t)}(n,\alpha)\) be the smallest \(m\) such that we can \(2\)-colour the edges of the complete \(t\)-uniform hypergraph on \(n\) vertices such that if \(X\subseteq [n]\) with \(\lvert X\rvert \geq m\) then there are at least \(\alpha \binom{\lvert X\rvert}{t}\) many \(t\)-subsets of \(X\) of each colour. For fixed \(n,t\) as we change \(\alpha\) from \(0\) to \(1/2\) does \(F^{(t)}(n,\alpha)\) increase continuously or are there jumps? Only one jump? Here growth is compared up to constant factors as \(n\to\infty\). At \(\alpha=0\), each color must occur, following the source's Ramsey convention.