The geometry and combinatorics of discrete line segment hypergraphs

Abstract

An $r$-segment hypergraph $H$ is a hypergraph whose edges consist of $r$ consecutive integer points on line segments in ${\mathbb{R}}^2$. In this paper, we bound the chromatic number $\chi (H)$ and covering number $\tau (H)$ of hypergraphs in this family, uncovering several interesting geometric properties in the process. We conjecture that for $r\ge 3$, the covering number $\tau (H)$ is at most $(r-1)\nu (H)$, where $\nu (H)$ denotes the matching number of $H$. We prove our conjecture in the case where $\nu (H)=1$, and provide improved (in fact, optimal) bounds on $\tau (H)$ for $r\le 5$. We also provide sharp bounds on the chromatic number $\chi (H)$ in terms of $r$, and use them to prove two fractional versions of our conjecture.