Comparing list-color functions of uniform hypergraphs with their chromatic polynomials (III)

For a hypergraph ${\cal H}$, 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 least non-negative integer $q$ such that $P({\cal H},k)=P_l({\cal H},k)$ holds for all integers $k\ge q$. In this article, we show that for any $r$-uniform hypergraph ${\cal H}$ of order $n$ and size $m$ and any $k$-assignment $L$ of ${\cal H}$, where $r\ge 3$, $P({\cal H},L)-P({\cal H},k)\ge \min \{0.02k, k-(m-1)\} k^{n-r-1}\sum_{e\in E({\cal H})} \left ( k-\left |\bigcap_{v\in e}L(v)\right | \right )$ holds for $k\ge m-1\ge 4$. It follows that $\tau'({\cal H})\le m-1$, improving the current best result on $\tau'({\cal H})$.