On the maximum number of isosceles right triangles in a finite point set

TL;DR

This paper investigates the maximum number of isosceles right triangles that can be formed from a finite set of points in the plane, providing exact counts for small sets and new bounds for larger sets.

Contribution

It offers exact solutions for small point sets and establishes new upper and lower bounds for the maximum number of such triangles in larger sets.

Findings

Exact counts for n ≤ 9 points

New upper bounds for larger n

New lower bounds for larger n

Abstract

Let $Q$ be a finite set of points in the plane. For any set $P$ of points in the plane, $S_{Q}(P)$ denotes the number of similar copies of $Q$ contained in $P$. For a fixed $n$, Erd\H{o}s and Purdy asked to determine the maximum possible value of $S_{Q}(P)$, denoted by $S_{Q}(n)$, over all sets $P$ of $n$ points in the plane. We consider this problem when $Q=\triangle$ is the set of vertices of an isosceles right triangle. We give exact solutions when $n\leq9$, and provide new upper and lower bounds for $S_{\triangle}(n)$.