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A special class of almost disjoint families

Published online by Cambridge University Press:  12 March 2014

Thomas E. Leathrum
Thomas E. Leathrum*
Affiliation:
Department of Mathematics and Computer Science, Dartmouth College, Hanover, New Hampshire 03755
*
Department of Mathematical Sciences, Berry College, Mt. Berry, Georgia 30149, E-mail: [email protected]
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Abstract

The collection of branches (maximal linearly ordered sets of nodes) of the tree <ωω (ordered by inclusion) forms an almost disjoint family (of sets of nodes). This family is not maximal — for example, any level of the tree is almost disjoint from all of the branches. How many sets must be added to the family of branches to make it maximal? This question leads to a series of definitions and results: a set of nodes is off-branch if it is almost disjoint from every branch in the tree; an off-branch family is an almost disjoint family of off-branch sets; and is the minimum cardinality of a maximal off-branch family.

Results concerning include: (in ZFC) , and (consistent with ZFC) is not equal to any of the standard small cardinal invariants or = 2ω. Most of these consistency results use standard forcing notions—for example, in the Cohen model.

Many interesting open questions remain, though—for example, whether .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 60 , Issue 3 , September 1995 , pp. 879 - 891
Copyright
Copyright © Association for Symbolic Logic 1995

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References

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