Multicolor list Ramsey numbers grow exponentially

Abstract

The list Ramsey number $R_{\ell }(H,k)$, recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list-coloring variant of the classical Ramsey number. They showed that if $H$ is a fixed $r$-uniform hypergraph that is not $r$-partite and the number of colors $k$ goes to infinity, $e^{\Omega (\sqrt{k})}\le R_{\ell }(H,k)\le e^{O(k)}$. We prove that $R_{\ell }(H,k)=e^{\Theta (k)}$ if and only if $H$ is not $r$-partite.

arXiv comment7 pages