Number of instances of patterns in finite subsets of the line and the plane

An instance of a polygon in a finite subset of the plane is represented by the vertices of the polygon. New upper bounds are proved on the number of instances of similar triangles and regular polygons in a finite subset of the plane. Interesting families of progressions in R are described. For any progression with commensurable pairwise distances between all points, a finite subset of the reals containing a maximal number of instances of the progression is contained in the integers.