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An order property for families of sets

Part of: Set theory

Published online by Cambridge University Press:  09 April 2009

N. H. Williams
N. H. Williams
Affiliation:
Department of MathematicsUniversity of QueenslandSt. Lucia, Queensland 4067, Australia
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Abstract

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We develop the idea of a θ-ordering (where θ is an infinite cardinal) for a family of infinite sets. A θ-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 θ-transversal, are simple consequences of A possessing a θ-ordering. Most of the paper is devoted to showing that under suitable restrictions, an almost disjoint family will have a θ-ordering. The restrictions involve either intersection conditions on A (the intersection of every λ-size subfamily of A has size at most κ) or a chain condition on A.

MSC classification

Secondary: 03E05: Other combinatorial set theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 3 , June 1988 , pp. 294 - 310
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Davies, R. O. and Erdös, P., ‘Splitting almost-disjoint collections of sets into subcollections admitting almost-transversals’, Infinite and finite sets, Vol. I, edited by Hajnal, A., Rado, R. and Sós, V. T., pp. 307–322 (North-Holland, 1975).Google Scholar
[2]Erdös, P. and Hajnal, A., ‘On a property of families of sets,’ Acta Math. Acad. Sci. Hungar. 12 (1961), 87–123.CrossRefGoogle Scholar
[3]Erdös, P. and Hachler, S. H., ‘On maximal almost-disjoint families over singular cardinals’, Infinite and finite sets, Vol. I, edited by Hajnal, A., Rado, R. and Sós, V. T., pp. 597–604 (North-Holland, 1975).Google Scholar
[4]Erdös, P., Milner, E. C. and Rado, R., ‘Intersection theorems for sustems of sets (III)’, J. Austral. Math. Soc. 18 (1974), 22–40.CrossRefGoogle Scholar
[5]Komjáth, P., ‘Families close to disjoint ones’, Acta Math. Acad. Sci. Hungar. 43 (1984), 199–207.CrossRefGoogle Scholar
[6]Williams, N. H., Combinatorial set theory (Studies in Logic and the Foundations of Mathematics, 91, North-Holland, 1977).CrossRefGoogle Scholar
[7]Williams, N. H., ‘Chain conditions and the existence of delta-families’, Math. Proc. Cambridge Philos. Soc. 83 (1978), 167–179.CrossRefGoogle Scholar
[8]Williams, N. H., ‘Families of sets having property B’, Period. Math. Hungar. 12 (1981), 181–190.CrossRefGoogle Scholar