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

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 $\Omega(n\log n)\leq S_{P}^{\nparallel}(n)\leq O(n^{3/2})$.