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

摘要

Let $Q$ be a finite set of points in the plane. For any set $P$ of points inthe 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 maximumpossible 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 ofvertices of an isosceles right triangle. We give exact solutions when $n\leq9$,and provide new upper and lower bounds for $S_{\triangle}(n)$.