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

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

2011-02-25

arxiv.org/abs/1102.5347v1

Mathematics

Combinatorics

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

Scientific paper

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 $S_{\triangle}(n)$.

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