On Ramsey numbers of hedgehogs

Abstract

The hedgehog $H_t$ is a 3-uniform hypergraph on vertices $1,\dots ,t+\left(\genfrac{}{}{0ex}{}{t}{2}\right)$ such that, for any pair $(i,j)$ with $1\le i<j\le t$, there exists a unique vertex $k>t$ such that $\{i,j,k\}$ is an edge. Conlon, Fox, and Rödl proved that the two-color Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-color Ramsey number grows exponentially in the number of its vertices. They asked whether the two-color Ramsey number of the hedgehog $H_t$ is nearly linear in the number of its vertices. We answer this question affirmatively, proving that $r(H_t)=O(t^2\ln t)$.

13 pages