Ramsey numbers for 1-degenerate 3-graphs

Abstract

We construct a 3-uniform 1-degenerate hypergraph on $n$ vertices whose 2-colour Ramsey number is $\hat{I}\text{c◯}(n^{3/2}/\log n)$. This shows that all remaining open cases of the hypergraph Burr-Erdős 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 $O\left(n^{3/2}\right)$.

6 pages, 2 figures; final version, to appear in Electronic Journal of Combinatorics