Erdős Problem #161

Canonical assertion

declared status 'open'. Formalized: no. Let $\alpha\in[0,1/2)$ and $n,t\geq 1$. Let $F^{(t)}(n,\alpha)$ be the largest $m$ 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? Current best: A conjecture of Erd\H{o}s, Hajnal, and Rado (see [562]) implies that$$F^{(t)}(n,0)\asymp \log_{t-1} n$$and results of Erd\H{o}s and Spencer imply that$$F^{(t)}(n,\alpha) \gg_\alpha (\log n)^{\frac{1}{t-1}}$$for all $\alpha>0$, and a similar upper bound holds for $\alpha$ close to $1/2$. Conlon, Fox, and Sudakov [CFS11] have proved that, for any fixed $\alpha>0$,$$F^{(3)}(n,\alpha) \ll_\alpha \sqrt{\log n}.$$Coupled with the lower bound above, this implies that there is only one jump for fixed $\alpha$ when $t=3$, at $\alpha=0$. For all $\alpha>0$ it is known that$$F^{(t)}(n,\alpha)\gg_t (\log n)^{c_\alpha}.$$See also [563]. Prize: $500. Tags: combinatorics, discrepancy, ramsey theory.