A Criterion for the Existence of Transversals of Set Systems

Nash‐Williams [ 6 ] formulated a condition that is necessary and sufficient for a countable family A=( A i ) i ∈ 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 ∩ i ∈ J A i =Ø for any uncountable J ⊆ I . In this paper, we formulate another criterion of a similar nature, and prove that it is equivalent to the criterion of Nash‐Williams for any family u. We also present a self‐contained proof that if ∩ i ∈ J A i =Ø for any uncountable J ⊆ I , then our condition is necessary and sufficient for the family u to have a transversal.