On Crossing-Families in Planar Point Sets

TL;DR

This paper improves bounds on the size of crossing families in planar point sets, showing that small point sets guarantee a crossing family of size 4, and providing upper bounds for larger sets.

Contribution

The paper presents new bounds on crossing families in planar point sets, improving previous results with bounds based on small cardinality sets.

Findings

Any set of at least 15 points contains a crossing family of size 4.

There exist point sets with no crossing family larger than approximately 8 times the ceiling of n/41.

Both results improve upon earlier known bounds.

Abstract

A $k$-crossing family in a point set $S$ in general position is a set of $k$ segments spanned by points of $S$ such that all $k$ segments mutually cross. In this short note we present two statements on crossing families which are based on sets of small cardinality: (1) Any set of at least 15 points contains a crossing family of size 4. (2) There are sets of $n$ points which do not contain a crossing family of size larger than $8\lceil \frac{n}{41} \rceil$. Both results improve the previously best known bounds.