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

antichain number
Cohen forcing
Hechler forcing
Matching number
Semidistributive forcing
Stick

ASJC Scopus subject areas

Geometry and Topology

Access to Document

10.1016/j.topol.2019.01.013

Cite this

@article{7e083bdf4913461c98eece4305619902,

title = "Antichains, the stick principle, and a matching number",

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.",

keywords = "antichain number, Cohen forcing, Hechler forcing, Matching number, Semidistributive forcing, Stick",

author = "William Chen and Geoff Galgon",

note = "Publisher Copyright: {\textcopyright} 2019 Elsevier B.V.",

year = "2019",

month = apr,

day = "1",

doi = "10.1016/j.topol.2019.01.013",

language = "English",

volume = "256",

pages = "73--85",

journal = "Topology and its Applications",

issn = "0166-8641",

publisher = "Elsevier B.V.",

}

TY - JOUR

T1 - Antichains, the stick principle, and a matching number

AU - Chen, William

AU - Galgon, Geoff

PY - 2019/4/1

Y1 - 2019/4/1

N2 - 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.

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 - antichain number

KW - Cohen forcing

KW - Hechler forcing

KW - Matching number

KW - Semidistributive forcing

KW - Stick

UR - https://www.scopus.com/pages/publications/85060995045

U2 - 10.1016/j.topol.2019.01.013

DO - 10.1016/j.topol.2019.01.013

M3 - Article

AN - SCOPUS:85060995045

SN - 0166-8641

VL - 256

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

Access to Document

Other files and links

Fingerprint

Cite this