Chain method for panchromatic colorings of hypergraphs

TL;DR

This paper establishes conditions under which hypergraphs can be colored with r colors so that every edge contains all colors, extending understanding of panchromatic colorings in hypergraph theory.

Contribution

It introduces a new chain method to prove the existence of panchromatic colorings for certain hypergraphs with bounded edge counts.

Findings

Proves existence of panchromatic colorings for hypergraphs with specific edge bounds.

Provides bounds on the number of edges for guaranteed panchromatic coloring.

Extends previous results to larger ranges of r and hypergraph sizes.

Abstract

We deal with an extremal problem concerning panchromatic colorings of hypergraphs. A vertex $r$-coloring of a hypergraph $H$ is \emph{panchromatic} if every edge meets every color. We prove that for every $3<r\leq\sqrt[3]{n/(100\ln n)}$, every $n$-uniform hypergraph $H$ with $|E(H)|\leq \frac{1}{20r^2}\left(\frac{n}{\ln n}\right)^{\frac {r-1}{r}}\left(\frac{r}{r-1}\right)^{n-1}$ has a panchromatic coloring with $r$ colors.