The geometry and combinatorics of discrete line segment hypergraphs

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.