Combinatorics of intervals in the plane I: trapezoids

TL;DR

This paper investigates the combinatorial structure of interval arrangements in the plane that form numerous trapezoids, revealing underlying algebraic properties and highlighting the role of degree 2 curves in these configurations.

Contribution

It characterizes the algebraic structure of interval sets forming many trapezoids and introduces new examples involving degree 2 curves.

Findings

Sets with many trapezoids have underlying algebraic structure.

Degree 2 curves play a key role in these configurations.

New examples of interval arrangements with abundant trapezoids are provided.

Abstract

We study arrangements of intervals in $\mathbb{R}^2$ for which many pairs form trapezoids. We show that any set of intervals forming many trapezoids must have underlying algebraic structure, which we characterise. This leads to some unexpected examples of sets of intervals forming many trapezoids, where an important role is played by degree 2 curves.