Submit progress

on Erdős #562 · graph theory / ramsey theory

The statement

Let $R_r(n)$ denote the $r$ -uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$ -colour all edges of the complete $r$ -uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$ -uniform hypergraph on $n$ vertices.

Prove that, for $r\geq 3$ ,

\log_{r-1} R_r(n) \asymp_r n,

where $\log_{r-1}$ denotes the $(r-1)$ -fold iterated logarithm. That is, does $R_r(n)$ grow like

2^{2^{\cdots n}}

where the tower of exponentials has height $r-1$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.