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

For a hypergraph $\mathcal{H}$, let $P(\mathcal{H},k)$ and $P_l(\mathcal{H},k)$ be its chromatic polynomial and list-color function respectively, and let ${\tau }^{\mathrm{\prime }}(\mathcal{H})$ be the least non-negative integer $q$ such that $P(\mathcal{H},k)=P_l(\mathcal{H},k)$ holds for all integers $k\ge q$. In this article, we show that for any $r$-uniform hypergraph $\mathcal{H}$ of order $n$ and size $m$ and any $k$-assignment $L$ of $\mathcal{H}$, where $r\ge 3$, $P(\mathcal{H},L)-P(\mathcal{H},k)\ge \min \{0.02k,k-(m-1)\}{k}^{n-r-1}{\sum }_{e\in E(\mathcal{H})}(k-\mid {\bigcap }_{v\in e}L(v)\mid )$ holds for $k\ge m-1\ge 4$. It follows that ${\tau }^{\mathrm{\prime }}(\mathcal{H})\le m-1$, improving the current best result on ${\tau }^{\mathrm{\prime }}(\mathcal{H})$.