
Bernardo M. Ábrego

Silvia FernándezMerchant

Daniel J. Katz

Levon Kolesnikov
Keywords:
Pattern, Similar copy, Similar triangle, Equilateral triangle, Arithmetic progression
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 $\lfloor{(4 n1)(n1)/18}\rfloor$. The number of $k$term arithmetic progressions that lie within an $n$point subset of the line is shown to be at most $(nr)(n+rk+1)/(2 k2)$, where $r$ is the remainder when $n$ is divided by $k1$. 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.