Antichains, the stick principle, and a matching number

William Chen; Geoff Galgon

doi:10.1016/j.topol.2019.01.013

Antichains, the stick principle, and a matching number

William Chen, Geoff Galgon

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate basic properties of three cardinal invariants involving ω₁: stick, antichain number, and matching number. The antichain number is the least cardinal κ for which there does not exist a subcollection of size κ of uncountable subsets of ω₁ with pairwise finite intersections and the matching number is the least cardinal κ for which there exists a subcollection X of size κ of order-type ω subsets of ω₁ so that every uncountable subset of ω₁ has infinite intersection with a member of X. We demonstrate how these numbers are affected by Cohen forcing and also prove some results about the effect of Hechler forcing. We also introduce a forcing notion to increase the matching number, and study its basic properties.

Original language	English
Pages (from-to)	73-85
Number of pages	13
Journal	Topology and its Applications
Volume	256
DOIs	https://doi.org/10.1016/j.topol.2019.01.013
State	Published - 1 Apr 2019

Keywords

Cohen forcing
Hechler forcing
Matching number
Semidistributive forcing
Stick
antichain number

ASJC Scopus subject areas

Geometry and Topology

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10.1016/j.topol.2019.01.013

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abstract = "We investigate basic properties of three cardinal invariants involving ω1: stick, antichain number, and matching number. The antichain number is the least cardinal κ for which there does not exist a subcollection of size κ of uncountable subsets of ω1 with pairwise finite intersections and the matching number is the least cardinal κ for which there exists a subcollection X of size κ of order-type ω subsets of ω1 so that every uncountable subset of ω1 has infinite intersection with a member of X. We demonstrate how these numbers are affected by Cohen forcing and also prove some results about the effect of Hechler forcing. We also introduce a forcing notion to increase the matching number, and study its basic properties.",

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AB - We investigate basic properties of three cardinal invariants involving ω1: stick, antichain number, and matching number. The antichain number is the least cardinal κ for which there does not exist a subcollection of size κ of uncountable subsets of ω1 with pairwise finite intersections and the matching number is the least cardinal κ for which there exists a subcollection X of size κ of order-type ω subsets of ω1 so that every uncountable subset of ω1 has infinite intersection with a member of X. We demonstrate how these numbers are affected by Cohen forcing and also prove some results about the effect of Hechler forcing. We also introduce a forcing notion to increase the matching number, and study its basic properties.

KW - Cohen forcing

KW - Hechler forcing

KW - Matching number

KW - Semidistributive forcing

KW - Stick

KW - antichain number

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SN - 0166-8641

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SP - 73

EP - 85

JO - Topology and its Applications

JF - Topology and its Applications

ER -

Antichains, the stick principle, and a matching number

Abstract

Keywords

ASJC Scopus subject areas

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