On crossing families of complete geometric graphs

TL;DR

This paper generalizes the concept of crossing families in complete geometric graphs and provides new results on their properties and sizes, extending the foundational work by Aronov et al. (1994).

Contribution

It introduces a broader concept of crossing families and establishes several new theoretical results related to their size and structure.

Findings

Generalized crossing family concept introduced

Proved lower bounds on crossing family sizes

Extended previous results by Aronov et al.

Abstract

A crossing family is a collection of pairwise crossing segments, this concept was introduced by Aronov et. al. (1994). They prove that any set of $n$ points (in general position) in the plain contains a crossing family of size $\sqrt{n/12}$. In this paper we present a generalization of the concept and give several results regarding this generalization.