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

Abstract

A problem originating with Erdős 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, Erdős and Szekeres. We give bounds on ${\Gamma }_k(n)$ for a general $k$ and large $n$.