Plane point sets with many squares or isosceles right triangles

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 $15\le n\le 50$.

31 pages, 9 tables, 10 figures