Asymptotic Bounds on the Equilateral Dimension of Hypercubes

Abstract

Abstract. A subset of the finite dimensional hypercube is said to be equilateral if the distance of any two distinct points equals a fixed value. The equilateral dimension of the hypercube is defined as the maximal size of its equilateral subsets. We study asymptotic bounds on the latter quantity considered as a function of two variables, namely dimension and distance.lorenz minder a , thomas sauerwald b and sven-ake wegner c IntroductionThe notion of equilateral sets, i.e., sets in which the distance of any two distinct points equals a fixed value, can be defined in broad generality -that is in arbitrary metric spaces.In 1983, Kusner [9] raised the question of determining equilateral sets in the finite dimensional ℓ p -spaces for 1 p ∞. He computed the so-called equilateral dimension, that is the maximal size of equilateral sets, of the Hilbert space ℓ 2 (n) and of ℓ ∞ (n). For the other cases of p he formulated conjectures which are very persuasive but turned out to constitute surprisingly hard and interdisciplinary problems. In particular, the case of ℓ 1 (n) appears to be a fairly easy exercise at a first glance but is resisting a complete solution for more than thirty years now. However, during this time many results, using techniques from various areas of mathematics such as functional analysis, probability theory, combinatorics, approximation theory and algebraic topology, have been obtained by several authors, e.g., Alon, Pudlák [1], Bandelt et al. [2] and Koolen et al. [12]. We refer to the survey [15] of Swanepoel for detailed references, an overview of the state of the art concerning equilateral sets in normed spaces and historical comments.A completely different area where equilateral sets can be studied are finite, undirected and connected graphs with the usual shortest-path metric. Being a structural invariant, it is a natural objective to compute the equilateral dimension of certain graph classes. In addition, equilateral sets might be of use in the context of information dissemination (similar to Feige et al. [7]), i.e., the problem of spreading a message held by a set of source nodes to a set of target nodes (broadcasting), by using an algorithm which regulates the communications in the neighborhood of any point. A third area in which equilateral sets occur naturally is the theory of codes. Equilateral subsets of the hypercube are nothing but equidistant codes in the language of coding theory (cf. the books of MacWilliams, Sloane [14] or Huffman, Pless [11]) and constitute a classical research topic with important applications for instance to error correcting codes. They are closely related to constant weight codes and to the theory of 2-designs.2010 Mathematics Subject Classification: Primary 05C12; Secondary 11H71, 51B20.