# List colourings of multipartite hypergraphs

## TL;DR

This paper investigates the list chromatic number of multipartite hypergraphs, extending known bounds, and shows that the container method provides optimal bounds for certain cases but not universally for all hypergraph sizes.

## Contribution

It extends bounds on list chromatic numbers to multipartite hypergraphs and analyzes the effectiveness of the container method across different hypergraph sizes.

## Key findings

- Container method yields optimal bounds for r=2 and r=3.
- For r≥4, the container method does not give optimal bounds.
- The function g(r,α) characterizes the list chromatic number in random hypergraphs.

## Abstract

Let $\chi_l(G)$ denote the list chromatic number of the $r$-uniform hypergraph~$G$. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if $G$ is simple and $d$-regular, then $\chi_l(G)\ge (1/(r-1)+o(1))\log_r d$. To see how close this inequality is to best possible, we examine $\chi_l(G)$ when $G$ is a random $r$-partite hypergraph with $n$ vertices in each class. The value when $r=2$ was determined by Alon and Krivelevich, here we show that $\chi_l(G)= (g(r,\alpha)+o(1))\log_r d$ almost surely, where $d$ is the expected average degree of~$G$ and $\alpha=\log_nd$. The function $g(r,\alpha)$ is defined in terms of "preference orders" and can be determined fairly explicitly. This is enough to show that the container method gives an optimal lower bound on $\chi_l(G)$ for $r=2$ and $r=3$, but, perhaps surprisingly, apparently not for $r\ge4$.