The deviation from right angles in $k$-subsets of points in the plane

TL;DR

This paper investigates how far points in the plane can deviate from forming right angles in k-subsets, providing bounds on the maximum deviation angle for various subset sizes.

Contribution

It introduces a relaxed measure of deviation from right angles in point sets and establishes bounds for this deviation across different subset sizes and point counts.

Findings

Bounds established for deviation angles in 4-point subsets: 4° to 9.292°.

Relation of the deviation measure to classical minimax angle problems for large n.

General bounds provided for deviation angles in k-subsets for arbitrary k and large n.

Abstract

A problem originating with Erd\H{o}s and Silverman in the 1970s asks for the minimum integer $r(k)$ such that any set of $n \ge r(k)$ points in the plane has some $k$-subset with no right angles. The case $k=4$ has an interesting gap between the known bounds, namely $8 \le r(4) \le 10$. Here, we consider a relaxation that quantifies the deviation from right angles. Specifically, we study $\Gamma_k(n)$, the supremum of angles $\gamma$ such that every $n$-set of points in $\mathbb{R}^2$ has a $k$-subset with all angles outside of the interval $90^\circ \pm \gamma$. We show that $4^\circ \le \Gamma_4(10) \le 9.292^\circ$. For large $n$, the quantity $\Gamma_3(n)$ is closely related to a classical minimax angle problem pioneered by Blumenthal, Erd\H{o}s and Szekeres. We give bounds on $\Gamma_k(n)$ for a general $k$ and large $n$.