On some cardinal functions related to quasi-uniformities

We show that if a topological space possesses two (distinct) compatible quasi-uniformities, then it admits at least 2(2 kappa)0 nontransitive quasi-uniformities. We also prove that if a quasi-uniform space (X, W) has a subspace A and an entourage W such that either {W(rc) :x is an element of A} or {W-1(x) :x is an element of A} does not have a subcollection of cardinality smaller than kappa covering A, then there are at least 2(2 kappa) quasi-uniformities belonging to the quasi-proximity class of W. (Here re is an infinite cardinal.) Finally we show that if the quasi-proximity class pi(W) of a quasi-uniformity W contains more than one quasi-uniformity and its coarsest member is transitive, then there are at least 2(2 kappa)0 transitive quasi-uniformities belonging to the quasi-proximity class pi(W).