Compare list-color functions of hypergraphs with their chromatic polynomials (I)
For any $r$-uniform hypergraph ${\cal H}=(V,E)$, where $r\ge 3$, let $P({\cal H},k)$ and $P_l({\cal H},k)$ be its chromatic polynomial and list-color function respectively, and let $\tau'({\cal H})$ be the smallest positive integer $q$ such that $P({\cal H},k)=P_l({\cal H},k)$ holds for all $k\ge q$. Let $\gamma({\cal H})$ denote the maximum value of $\gamma_{e}({\cal H})$ over all edges $e$ in ${\cal H}$, where $\gamma_{e}({\cal H})$ is the number of edges $e'\in E$ such that $|e\cap e'|=r-1$. In this article, we show that $\tau'({\cal H})\le |E|-1$ if $2\le |E|\le 4$, and $\tau'({\cal H})\le 0.6(|E|-1) +0.5\gamma({\cal H})\le |E|-1$ if $|E|\ge 5$ and $\gamma({\cal H})\le 0.8(|E|-1)$. Our result improves the latest known result by Wang, Qian and Yan that $\tau'({\cal H})\le 1.1346(|E|-1)$.