Vol. 4, No. 1, 2011

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

Bernardo M. Ábrego, Silvia Fernández-Merchant and David B. Roberts

Vol. 4 (2011), No. 1, 27–42
Abstract

Let $Q$ be a finite set of points in the plane. For any set $P$ of points in the plane, ${S}_{Q}\left(P\right)$ denotes the number of similar copies of $Q$ contained in $P$. For a fixed $n$, Erdős and Purdy asked for the maximum possible value of ${S}_{Q}\left(P\right)$, denoted by ${S}_{Q}\left(n\right)$, 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 ${S}_{△}\left(n\right)$.

Keywords
Erdős problems, similar triangles, isosceles right triangles
Primary: 52C10
Secondary: 05C35