Point-sets in general position with many similar copies of a pattern

TL;DR

This paper investigates the maximum number of similar copies of a pattern within large point sets in the plane, providing new bounds using geometric constructions and analyzing special cases like parallelogram-free sets.

Contribution

It introduces a general construction method for lower bounds on similar pattern copies and improves bounds for specific patterns like triangles and polygons.

Findings

New lower bounds for $S_{P}(n,m)$ using Minkowski sums.

Enhanced bounds for triangles and regular polygons.

Bounds for parallelogram-free sets: $ ext{Omega}(n ext{log} n)$ to $O(n^{3/2})$.

Abstract

For every pattern $P$, consisting of a finite set of points in the plane, $S_{P}(n,m)$ is defined as the largest number of similar copies of $P$ among sets of $n$ points in the plane without $m$ points on a line. A general construction, based on iterated Minkovski sums, is used to obtain new lower bounds for $S_{P}(n,m)$ when $P$ is an arbitrary pattern. Improved bounds are obtained when $P$ is a triangle or a regular polygon with few sides. It is also shown that $S_{P}(n,m)\geq n^{2-\epsilon}$ whenever $m(n)\to \infty$ as $n \to\infty$. Finite sets with no collinear triples and not containing the 4 vertices of any parallelogram are called \emph{parallelogram-free}. The more restricted function $S_{P} ^{\nparallel}(n)$, defined as the maximum number of similar copies of $P$ among parallelogram-free sets of $n$ points, is also studied. It is proved that $\Omega(n\log n)\leq S_{P}^{\nparallel}(n)\leq O(n^{3/2})$.