On the number of similar instances of a pattern in a finite set

Bernardo M. Ábrego¹ ; Silvia Fernández-Merchant¹ ; Daniel J. Katz¹ ; Levon Kolesnikov¹

¹California State University, Northridge

The electronic journal of combinatorics, Tome 23 (2016) no. 4

Résumé

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.

Détail

DOI : 10.37236/4972

Classification : 05B25, 05A99, 52C10
Mots-clés : pattern, similar copy, similar triangle, equilateral triangle, arithmetic progression