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A Criterion for the Existence of Transversals of Set Systems

Authors:	Wojciechowski Jerzy

Affiliation:	Department of Mathematics, West Virginia University P.O. Box 6310, Morgantown, West Virginia 26506-6310, USA. E-mail: jerzy{at}math.wvu.edu

Abstract:	Nash-Williams [6] formulated a condition that is necessary andsufficient for a countable family A=(A_i)_iI of sets to have atransversal. In [7] he proved that his criterion applies alsowhen we allow the set I to be arbitrary and require only that ${cap}$ _iJA_i=Ø for any uncountable J ${subseteq}$ I. In this paper, we formulateanother criterion of a similar nature, and prove that it isequivalent to the criterion of Nash-Williams for any familyu. We also present a self-contained proof that if ${cap}$ _iJA_i=Øfor any uncountable J ${subseteq}$ I, then our condition is necessary andsufficient for the family u to have a transversal.

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