Hypergraph Ramsey numbers

TL;DR

This paper advances bounds on hypergraph Ramsey numbers, providing new upper and lower estimates for 3-uniform hypergraphs, including the first superexponential lower bound for fixed s, and improving understanding of 3-color Ramsey numbers.

Contribution

The paper introduces improved upper bounds for r_k(s,n) with fixed s, establishes the first superexponential lower bounds for r_3(s,n), and refines estimates for r_3(n,n,n), advancing hypergraph Ramsey theory.

Findings

r_3(s,n)  2^{n^{s-2}\u221an} (improved upper bound)

r_3(s,n)  2^{c_1 s n f(n/s)} (new lower bound)

r_3(n,n,n) 2^{n^{c f n}} (improved lower bound)

Abstract

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 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 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}/ polylog n the exponent of the previous upper bound of Erdos and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there are constants c_1,c_2>0 such that r_3(s,n) \geq 2^{c_1 sn \log (n/s)} for all 4 \leq s \leq c_2n. When s is a constant, it gives the first superexponential lower bound for r_3(s,n), answering an open question posed by Erdos 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 Erdos 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.