A moore bound for simplicial complexes

Let X be a d-dimensional simplicial complex with N faces of dimension d-1. Suppose that every (d-1)-face of X is contained in at least κ ≥ d + 2 faces of X, of dimension d. Extending the classical Moore bound for graphs, it is shown that X must contain a ball B of radius at most |"log _κ-d N] whose d-dimensional homology H_d(B) is non-zero. The Ramanujan complexes constructed by Lubotzky, Samuels and Vishne are used to show that this upper bound on the radius of B cannot be improved by more than a multiplicative constant factor.