Minimal abundant packings and choosability with separation

TL;DR

This paper advances the understanding of abundant packings in combinatorics by providing new bounds and asymptotic results, and applies these findings to graph list coloring with separation constraints.

Contribution

It offers improved estimates for maximum packings, determines the minimal parameters for their existence, and solves an open problem on list coloring limits in graphs.

Findings

Improved bounds for $P(v,k,t)$ near the critical range.

Asymptotic determination of the minimal $v$ for abundant packings.

Proved the existence of the limit of $rac{ ext{chi}_ ext{ell}(K_n,c)}{ oot{c}{n}}$ and found exact values for infinitely many $n$.

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 $n$.