CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

In this talk we consider the following class of problems in Discrete Geometry. Let $P$ be a finite point-set, called the pattern, in the $d$-dimensional euclidean space. A set $Q$ is considered a copy of $P$, if $Q$ is the image of the pattern $P$ under a specified geometric transformation. The problem consists of maximizing the number of copies of $P$ over all sets $Q$ with $n$ elements. There are different versions of this problem depending on the pattern and on the allowed geometric transformations. Some of the more common transformations are translations, rotations, reflections, and dilations. This class of problems was proposed by Paul Erdős and George Purdy in the 1970s and most of the variants are still open problems. In this talk we will present recent results.