On The Number of Similar Instances of a Pattern in a Finite Set
| Autor: | Abrego, Bernardo, Fernandez-Merchant, Silvia, Katz, Daniel J., Kolesnikov, Levon |
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| Rok vydání: | 2014 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 24 pages |
| Databáze: | arXiv |
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