Coloring Hasse diagrams and disjointness graphs of curves

Abstract

Given a family of curves $\mathcal{C}$ in the plane, its disjointness graph is the graph whose vertices correspond to the elements of $\mathcal{C}$, and two vertices are joined by an edge if and only if the corresponding sets are disjoint. We prove that for every positive integer $r$ and $n$, there exists a family of $n$ curves whose disjointness graph has girth $r$ and chromatic number $\Omega(\frac{1}{r}\log n)$. In the process we slightly improve Bollob\'as's old result on Hasse diagrams and show that our improved bound is best possible for uniquely generated partial orders.