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