Silver Cubes

An n × n matrix A is said to be silver if, for i = 1,2,...,n, each symbol in {1,2,...,2n − 1} appears either in the ith row or the ith column of A. The 38th International Mathematical Olympiad asked whether a silver matrix exists with n = 1997. More generally, a silver cube is a triple (K _n ^d , I, c) where I is a maximum independent set in a Cartesian power of the complete graph K _n , and is a vertex colouring where, for v ∈ I, the closed neighbourhood N[v] sees every colour. Silver cubes are related to codes, dominating sets, and those with n a prime power are also related to finite geometry. We present here algebraic constructions, small examples, and a product construction. The nonexistence of silver cubes for d = 2 and some values of n, is proved using bounds from coding theory. Luis A. Goddyn: This research was supported by a Canada NSERC Discovery Grant. Ebadollah S. Mahmoodian: Partially supported by the institutes CECM and IRMACS and the departments of Mathematics and Computing Science at Simon Fraser University. Grateful thanks are extended here and also to the Institute for Advanced Studies in Basic Sciences, Iran for support in the final stages of this paper.