Hypergraph Ramsey numbers

The Ramsey number rk(s, n) is the minimum N such that every red-blue coloring of the k-tuples of an N -element set contains either 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 rk(s, n) for k ≥ 3 and s fixed. In particular, we show that r3(s, n) ≤ 2 n log , which improves by a factor of n/polylogn 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 are constants c1, c2 > 0 such that r3(s, n) ≥ 2 c1 sn log(n/s) for all 4 ≤ s ≤ c2n. When s is a constant, it gives the first superexponential lower bound for r3(s, n), answering an open question posed by Erdős and Hajnal in 1972. Next, we consider the 3color Ramsey number r3(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 r3(n, n, n) ≥ 2 nc log n . Finally, we make some progress on related hypergraph Ramsey-type problems.