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A CRITERION FOR THE EXISTENCE OF TRANSVERSALS OF SET SYSTEMS

Published online by Cambridge University Press:  01 December 1997

JERZY WOJCIECHOWSKI
JERZY WOJCIECHOWSKI
Affiliation:
Department of Mathematics, West Virginia University, P.O. Box 6310, Morgantown, West Virginia 26506-6310, USA. E-mail: [email protected]
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Abstract

Nash-Williams [6] formulated a condition that is necessary and sufficient for a countable family [Ascr ]=(Ai)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 [xcap ]i∈JAi=[emptyv ] 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 [ufr ]. We also present a self-contained proof that if [xcap ]i∈JAi=[emptyv ] for any uncountable J⊆I, then our condition is necessary and sufficient for the family [ufr ] to have a transversal.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 56 , Issue 3 , December 1997 , pp. 491 - 503
Copyright
The London Mathematical Society 1997

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