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 \(\lfloor{(4 n-1)(n-1)/18}\rfloor\). 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.