On The Number of Similar Instances of a Pattern in a Finite Set

TL;DR

This paper establishes new bounds on the number of similar patterns, like equilateral triangles and arithmetic progressions, within finite point sets in the line and plane, using a novel ordering relations method.

Contribution

It introduces a new general method based on ordering relations to derive bounds on pattern counts in finite sets, including classifications of optimal configurations.

Findings

Maximum of n-1)(n-1)/18 equilateral triangles in n-point plane sets.

Upper bound for k-term arithmetic progressions in n-point line sets.

Full classification of configurations achieving these bounds.

Abstract

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is shown to be no more than $\lfloor{(4 n-1)(n-1)/18}\rfloor$. The number of $k$-term arithmetic progressions that lie within an $n$-point subset of the line is shown to be at most $(n-r)(n+r-k+1)/(2 k-2)$, where $r$ is the remainder when $n$ is divided by $k-1$. This upper bound is achieved when the $n$ points themselves form an arithmetic progression, but for some values of $k$ and $n$, it can also be achieved for other configurations of the $n$ points, and a full classification of such optimal configurations is given. These results are achieved using a new general method based on ordering relations.