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

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 ⌊(4n− 1)(n− 1)/18⌋. 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)/(2k − 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.