Minimal abundant packings and choosability with separation

Abstract

A (v, k, t) packing of size b is a system of b subsets (blocks) of a v-element underlying set such that each block has k elements and every t-set is contained in at most one block. P(v, k, t) stands for the maximum possible b. A packing is called abundant if \(b> v\). We give new estimates for P(v, k, t) around the critical range, slightly improving the Johnson bound and asymptotically determine the minimum \(v=v_0(k,t)\) when abundant packings exist. For a graph G and a positive integer c, let \(\chi _\ell (G,c)\) be the minimum value of k such that one can properly color the vertices of G from any assignment of lists L(v) such that \(|L(v)|=k\) for all \(v\in V(G)\) and \(|L(u)\cap L(v)|\le c\) for all \(uv\in E(G)\). Kratochvíl, Tuza and Voigt in 1998 asked to determine \(\lim _{n\rightarrow \infty } \chi _\ell (K_n,c)/\sqrt{cn}\) (if it exists). Using our bound on \(v_0(k,t)\), we prove that the limit exists and equals 1. Given c, we find the exact value of \(\chi _\ell (K_n,c)\) for infinitely many n.