Hypergraph Ramsey numbers with quasipolynomial growth rate

TL;DR

This paper constructs specific 3-uniform hypergraphs with quasipolynomial Ramsey numbers, demonstrating the first known examples that are neither polynomial nor exponential in growth rate.

Contribution

It introduces the first known hypergraphs with quasipolynomial Ramsey numbers, expanding understanding of growth rates in hypergraph Ramsey theory.

Findings

Constructed a 3-graph with $r(H_2,n)=n^{ ext{Theta}( ext{log} n)}$

Generalized to a family of 3-graphs with similar quasipolynomial growth

First examples of hypergraph Ramsey numbers neither polynomial nor exponential

Abstract

For a 3-uniform hypergraph (3-graph) $F$, let $r(F,n)$ be the smallest $N$ such that any $N$-vertex $F$-free 3-graph has an independent set of size $n$. We construct a $3$-graph $H_2$ with six vertices and five edges such that $r(H_2,n)=n^{\Theta(\log n)}$, and a more general family of $3$-graphs $F$ for which $r(F,n)=n^{\log^{\Theta(1)}(n)}$. These are the first examples of such Ramsey number known to be neither polynomial nor exponential.