A survey of hypergraph Ramsey problems

TL;DR

This survey reviews recent developments and open problems in hypergraph Ramsey theory, focusing on the classical hypergraph Ramsey numbers and their combinatorial properties.

Contribution

It provides a comprehensive overview of recent research and open questions related to hypergraph Ramsey numbers and their growth rates.

Findings

Summary of known bounds on hypergraph Ramsey numbers

Identification of key open problems in the field

Discussion of recent techniques and results

Abstract

The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We survey a variety of problems and results in hypergraph Ramsey theory that have grown out of understanding the quantitative aspects of $r_k(s,n)$. Our focus is on recent developments and open problems.