A CRITERION FOR THE EXISTENCE OF TRANSVERSALS OF SET SYSTEMS

Nash-Williams [6] formulated a condition that is necessary and suffici ent for a countable family A = (A(i))(i is an element of I) of sets to have a transversal. In [7] he proved that his criterion applies also when we allow the set I to be arbitrary and require only that boolean AND(i is an element of J) A(i) = empty set for any uncountable J subse t of or equal to I. In this paper, we formulate another criterion of a similar nature, and prove that it is equivalent to the criterion of N ash-Williams for any family U. We also present a self-contained proof that if boolean AND(i is an element of J) A(i) = empty set for any unc ountable J subset of or equal to I, then our condition is necessary an d sufficient for the family U to have a transversal.