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 $S_P(n,m)\ge n^{2-\epsilon }$ 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 $\Omega (n\log n)\le S_P^{\nparallel }(n)\le O(n^{3/2})$.

May 3 version. 21 pages, 10 figures