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

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 $n$, Erdős 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 $Q=△$ is the set of vertices of an isosceles right triangle. We give exact solutions when $n\le 9$, and provide new upper and lower bounds for $S_{△}(n)$.

arXiv comment15 pages, 6 figures. Research supported in part by CURM, BYU, and NSF(Award \#: DMS - 0636648)