1 research outputs found
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 -point subset of the
plane is shown to be no more than . The number
of -term arithmetic progressions that lie within an -point subset of the
line is shown to be at most , where is the
remainder when is divided by . This upper bound is achieved when the
points themselves form an arithmetic progression, but for some values of
and , it can also be achieved for other configurations of the
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 page