Ramsey numbers for 1-degenerate 3-graphs

TL;DR

This paper constructs a 3-uniform 1-degenerate hypergraph demonstrating large 2-color Ramsey numbers, disproving remaining cases of a conjecture, and establishes near-sharp upper bounds for generalized hedgehogs.

Contribution

It provides the first explicit construction of hypergraphs with large Ramsey numbers and refutes open cases of the hypergraph Burr-Erdős conjecture.

Findings

Constructed a hypergraph with Ramsey number Ω(n^{3/2}/log n)

Disproved remaining open cases of the Burr-Erdős conjecture

Proved upper bounds of O(n^{3/2}) for generalized hedgehogs

Abstract

We construct a 3-uniform 1-degenerate hypergraph on $n$ vertices whose 2-colour Ramsey number is $\Omega\big(n^{3/2}/\log n\big)$. This shows that all remaining open cases of the hypergraph Burr-Erd\H{o}s conjecture are false. Our graph is a variant of the celebrated hedgehog graph. We additionally show near-sharp upper bounds, proving that all 3-uniform generalised hedgehogs have 2-colour Ramsey number $O\big(n^{3/2}\big)$.