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

For any r-uniform hypergraph H with m (≥ 2) edges, let P(H, k) and P_l(H, k) be the chromatic polynomial and the list-color function of H respectively, and let ρ(H) denote the minimum value of |e \ e′| among all pairs of distinct edges e, e′ in H. We will show that if r ≥ 3, ρ(H) ≥ 2 and m ≥ ρ(H)³/2 +1, then P_l(H, k) = P(H, k) holds for all integers k ≥ 2.4(m−1) / ρ(H) log(m−1).