Point-sets with Many Similar Copies of a Pattern: without a Fixed Number of Collinear Points or in Parallelogram-free Position

Let P be a finite pattern, that is, a finite set of points in the plane. We consider the problem of maximizing the number of similar copies of P over all sets of n points in the plane under two general position restrictions: (1) Over all sets of n points with no m points on a line. We call this maximum SP (n,m). (2) Over all sets of n points with no collinear triples and not containing the 4 vertices of any parallelogram. These sets are called parallelogram-free and the maximum is denoted by S∦ P (n). We prove that SP (n,m) ≥ n2−ε whenever m(n) → ∞ as n →∞ and that Ω(n log n) ≤ S∦ P (n) ≤ O(n).