A positive fraction mutually avoiding sets theorem

TL;DR

This paper extends a classical geometric theorem by proving a fractional version, demonstrating that large point sets contain multiple mutually avoiding subsets with guaranteed size bounds, applicable in higher dimensions.

Contribution

It introduces a fractional version of the mutually avoiding sets theorem, establishing the existence of multiple large mutually avoiding subsets in large point sets, with explicit size bounds.

Findings

Existence of large mutually avoiding subsets in large point sets.

Fractional version with size bounds of (1/k^4) for subsets.

Applicability of results in higher dimensions.

Abstract

Two sets $A$ and $B$ of points in the plane are \emph{mutually avoiding} if no line generated by any two points in $A$ intersects the convex hull of $B$, and vice versa. In 1994, Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach, and Schulman showed that every set of $n$ points in the plane in general position contains a pair of mutually avoiding sets each of size at least $\sqrt{n/12}$. As a corollary, their result implies that for every set of $n$ points in the plane in general position one can find at least $\sqrt{n/12}$ segments, each joining two of the points, such that these segments are pairwise crossing. In this note, we prove a fractional version of their theorem: for every $k > 0$ there is a constant $\varepsilon_k > 0$ such that any sufficiently large point set $P$ in the plane contains $2k$ subsets $A_1,\ldots, A_{k},B_1,\ldots, B_k$, each of size at least $\varepsilon_k|P|$, such that every pair of sets $A = \{a_1,\ldots, a_k\}$ and $B = \{b_1,\ldots, b_k\}$, with $a_i \in A_i$ and $b_i \in B_i$, are mutually avoiding. Moreover, we show that $\varepsilon_k = \Omega(1/k^4)$. Similar results are obtained in higher dimensions