Transversals and Multitransversals

A transversal of a family .Fof sets is a family of pairwise distinct elements, one from each member of .y , and a multitransversal of .IF is a family of pairwise disjoint subsets, one of each member of F . The main result of this note, Theorem 4, gives necessary and sufficient conditions on families sa/ and .4 of cardinals in order that every family J whose members have cardinals given by sl should have (i) a transversal, (ii) a multitransversal whose members have cardinals given by .4. Our conditions turn out to involve the notion of a weakly inaccessible cardinal and that of a stationary set of ordinals . Our result (announced in [1]) amounts to saying that the test families .y, whose "good behaviour" implies that of every other family with the same cardinalities, are those whose members are sets of the form Ix : x < A}, where AA is an ordinal.