J Austral Math Soc Ser A 44 pp294--310, 1988.

An Order Property for Families of Sets

N. H. Williams

Abstract

We develop the idea of a q-ordering (where q is an infinite cardinal) for a family of infinite sets. A q-ordering of the family A is a well ordering of A which decomposes A into a union of pairwise disjoint intervals in a special way, which facilitates certain transfinite constructions. We show that several standard combinatorial properties, for instance that of the family A having a q-transversal, are simple consequences of A possessing a q-ordering. Most of the paper is devoted to showing that under suitable restrictions, an almost disjoint family will have a q-ordering. The restrictions involve either intersection conditions on A (the intersection of every l-size subfamily of A has size at most k) or a chain condition on A.

1980 AMS Subject Classification: 03E05, 04A20

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Authors

N. H. Williams

Department of Mathematics, University of Queensland, St. Lucia, Queensland 4067, Australia.