Families of sets having property B

Williams, N. H.

doi:10.1007/bf01849785

Cited by 1 publication

(3 citation statements)

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“…Combining Theorems 3.3 and 4.3, together with the observation that for regular K, every almost disjoint («, /c)-family has a /c-transversal provides a proof of ( [8], Theorem 3.2) (which can be extended by using Theorem 3.4 as well Put T = {xp; 0 < K}, SO always \Bp n T| > 1. We claim that \B D T| < 9 for each B € B, so T is a 0 + -transversal of B.…”

Section: Theorem 43 Le< a Be A (A /C)-family With A 0-ordering Anmentioning

confidence: 90%

“…Suppose \S\ = n and let A be a family of subsets of S satsifying C(n + ,0). Then \A\ < y, provided either (8) 0+ <fi and VA € A(\A\ > 0), or (9) 0 < n and cf{0) ^ ef(ji) and VA G A{\A\ > 0).…”

Section: Intersection Conditionsmentioning

confidence: 99%

“…terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700032110 [8] An order property for families of sets 301…”

Section: Intersection Conditionsmentioning

confidence: 99%

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An order property for families of sets

Williams

1988

J Aust Math Soc A

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We develop the idea of a ^-ordering (where 6 is an infinite cardinal) for a family of infinite sets. A 0-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 ^-transversal, are simple consequences of A possessing a 0-ordering. Most of the paper is devoted to showing that under suitable restrictions, an almost disjoint family will have a 0-ordering. The restrictions involve either intersection conditions on A (the intersection of every A-size subfamily of A has size at most K) or a chain condition on A.

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Section: Theorem 43 Le< a Be A (A /C)-family With A 0-ordering Anmentioning

confidence: 90%

Section: Intersection Conditionsmentioning

confidence: 99%