Edge-coloring linear hypergraphs with medium-sized edges

TL;DR

This paper investigates list edge coloring of linear hypergraphs with medium-sized edges, providing bounds that are optimal up to a constant factor, motivated by the Erdős-Faber-Lovász conjecture.

Contribution

It establishes new bounds for list edge coloring in linear hypergraphs with medium-sized edges, extending understanding related to the EFL conjecture.

Findings

List edge coloring bounds are proportional to n/(i-1)

Bounds are optimal up to a constant factor

Applicable for hyperedges between sizes i and C_{i,ε}√n

Abstract

Motivated by the Erd\H{o}s-Faber-Lov\'{a}sz (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We show that if the hyper-edge sizes are bounded between $i$ and $C_{i,\epsilon} \sqrt{n}$ inclusive, then there is a list edge coloring using $(1 + \epsilon) \frac{n}{i - 1}$ colors. The dependence on $n$ in the upper bound is optimal (up to the value of $C_{i,\epsilon}$).