Coloring Hasse diagrams and disjointness graphs of curves

TL;DR

This paper explores the coloring properties of disjointness graphs of plane curves, establishing bounds on girth and chromatic number, and improves classical results on Hasse diagrams with optimal bounds for certain partial orders.

Contribution

It introduces new bounds on the chromatic number of disjointness graphs with specified girth and refines classical results on Hasse diagrams, showing these bounds are tight for certain partial orders.

Findings

Existence of curve families with specified girth and logarithmic chromatic number

Improved bounds on Hasse diagram coloring properties

Optimal bounds for uniquely generated partial orders

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.