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 $\Omega\big(n^{3/2}/\log n\big)$. 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 $O\big(n^{3/2}\big)$.