Hypergraph Ramsey numbers

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that every red-blue coloring of the $k$-tuples of an $N$-element set contains a red set of size $s$ or a blue set of size $n$, where a set is called red (blue) if all $k$-tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for $r_k(s,n)$ for $k \geq 3$ and $s$ fixed. In particular, we show that

$$r_3(s,n) \leq 2^{n^{s-2}\log n},$$

which improves by a factor of $n^{s-2}/\textrm{polylog} n$ the exponent of the previous upper bound of Erdős and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there is a constant $c>0$ such that

$$r_3(s,n) \geq 2^{c sn \log (\frac{n}{s}+1)}$$

for all $4 \leq s \leq n$. For constant $s$, this gives the first superexponential lower bound for $r_3(s,n)$, answering an open question posed by Erdős and Hajnal in 1972. Next, we consider the $3$-color Ramsey number $r_3(n,n,n)$, which is the minimum $N$ such that every $3$-coloring of the triples of an $N$-element set contains a monochromatic set of size $n$. Improving another old result of Erdős and Hajnal, we show that

$$r_3(n,n,n) \geq 2^{n^{c \log n}}.$$

Finally, we make some progress on related hypergraph Ramsey-type problems.