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AG Audience

English

30 Minutes

Saarbrücken

Given a metric space M and a constant d>0, a subset of M is said to be d-equidistant if the distance of any two distinct points in this set is equal to d. It is a natural task to compute the maximal size of d-equidistant sets, sometimes refered to as the (d-)equilateral dimension, of M or at least to find upper and lower bounds for the latter. For example, it is easy to see that the equilateral dimension of the space l^1(n) is greater or equal to 2n but it is a long outstanding problem to show that the equality is valid. The best upper bound known so far, cn log n for some constant c, is due to Alon and Pudlak.

In the talk we study the equilateral dimension of the n-dimensional hypercube endowed with the usual Hamming distance. Amongst other results we derive asymptotic bounds on the equilateral dimension where we consider d as a function of n and let n tend to infinity. For instance, we are able to show that the equilateral dimension grows asymptotically at least as fast as n^(2/3) for d=n^a for 0<a<1 and that the minimal asymptotic growth is matched exactly for d=n^(1/3).

The results presented in this talk are joint work with L. Minder and T. Sauerwald.

Thomas Sauerwald

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