Plane point sets with many squares or isosceles right triangles

TL;DR

This paper investigates the maximum number of squares and isosceles right triangles that can be formed by n points in the plane, providing exact counts for small n and bounds for larger n, along with some structural insights.

Contribution

It determines exact maximum counts for small n and establishes lower bounds for larger n for both squares and isosceles right triangles, advancing understanding of geometric configurations.

Findings

Exact values for maximum squares for n ≤ 17.

Exact values for maximum isosceles right triangles for n ≤ 14.

Lower bounds for larger n for both shapes.

Abstract

How many squares are spanned by $n$ points in the plane? Here we study the corresponding maximum possible number $S_{\square}(n)$ of squares and determine the exact values for all $n\le 17$. For $18\le n\le 100$ we give lower bounds for $S_{\square}(n)$. Besides that a few preliminary structural results are obtained. For the related problem of the maximum possible number of isosceles right triangles we determine the exact values for $n\le 14$ and give lower bounds for $15\le n\le 50$.