Geometric Sidon Problems

Abstract

This paper considers geometric problems of the following type: given a point set $P\subset {\mathbb{R}}^2$, one seeks a large subset avoiding a prescribed geometric configuration. Our main result states that, for any $P\subset {\mathbb{R}}^2$, there exists a subset $P^{\prime }\subset P$ with $|P^{\prime }|\gg |P{|}^{1/3}$ such that all of the distances determined by $P^{\prime }$ are distinct. This improves a result of Charalambides. We make heavy use of a result of Li and Postle concerning the independence number of hypergraphs which satisfy some edge distribution conditions, as well as tools from incidence geometry.