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FORCING AND THE HALPERN–LÄUCHLI THEOREM

Published online by Cambridge University Press:  09 September 2019

NATASHA DOBRINEN  and
DANIEL HATHAWAY
NATASHA DOBRINEN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF DENVER C.M. KNUDSON HALL, ROOM 300 2390 S. YORK ST. DENVER, CO80208, USA E-mail: natasha.dobrinen@du.eduURL:http://web.cs.du.edu/∼ndobrine
DANIEL HATHAWAY
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF VERMONT 82 UNIVERSITY PLACE. BURLINGTON, VT05401, USA E-mail: daniel.hathaway@uvm.eduURL: http://mysite.du.edu/∼dhathaw2/
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Abstract

We investigate the effects of various forcings on several forms of the Halpern– Läuchli theorem. For inaccessible κ, we show they are preserved by forcings of size less than κ. Combining this with work of Zhang in [17] yields that the polarized partition relations associated with finite products of the κ-rationals are preserved by all forcings of size less than κ over models satisfying the Halpern– Läuchli theorem at κ. We also show that the Halpern–Läuchli theorem is preserved by <κ-closed forcings assuming κ is measurable, following some observed reflection properties.

Keywords

03E0203E1003E3503E5503E5705D10forcinggeneric absolutenessinfinitary combinatoricspartition relationsRamsey theorylarge cardinals

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 87 - 102
Copyright
Copyright © The Association for Symbolic Logic 2019 

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