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 $|L(u)\cap L(v)|\leq c$ for all $uv\in E(G)$. Kratochv\'{\i}l, Tuza and Voigt in 1998 asked to determine $\lim_{n\rightarrow \infty} \chi_\ell(K_n,c)/\sqrt{cn}$ (if 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 $n$.