Mixing properties and the chromatic number of ramanujan complexes

Ramanujan complexes are high-dimensional simplicial complexes generalizing Ramanujan graphs. A result of Oh on quantitative property (T) for Lie groups over local fields is used to deduce a Mixing Lemma for such complexes. As an application, we prove that nonpartite Ramanujan complexes have "high girth" and high-chromatic number, generalizing a well-known result about Ramanujan graphs.