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Transactions of the American Mathematical Society

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Nonhomogeneous analytic families of trees


Author: James Hirschorn
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 03E15; Secondary 03E40, 05D40, 28A12
DOI: https://doi.org/10.1090/tran/7517
Published electronically: June 10, 2019
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Abstract: We consider a dichotomy for analytic families of subtrees of a tree $ \mathbb{T}$ stating that either there is a colouring of the nodes of $ \mathbb{T}$ for which all but finitely many levels of every tree in the family are nonhomogeneous or else the family contains an uncountable antichain. This dichotomy implies that every nontrivial Souslin poset satisfying the countable chain condition adds a splitting real.

We then reduce the dichotomy to a conjecture of Sperner theory. This conjecture concerns the asymptotic behaviour of the product of the sizes of the $ m$-shades of pairs of cross-$ t$-intersecting families.


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Additional Information

James Hirschorn
Affiliation: Quantitative Technologies (Canada), Toronto, Ontario, Canada
Email: James.Hirschorn@quantitative-technologies.com

DOI: https://doi.org/10.1090/tran/7517
Keywords: Splitting real, Souslin forcing, cross-$t$-intersecting families
Received by editor(s): August 12, 2008
Received by editor(s) in revised form: September 4, 2017
Published electronically: June 10, 2019
Additional Notes: This research was primarily supported by Lise Meitner Fellowship, Fonds zur Förderung der wissenschaftlichen Forschung, Project No. M749-N05; the first version was completed with partial support of Consorcio Centro de Investigación Matemática, Spanish Government grant No. SB2002-0099.
Article copyright: © Copyright 2019 American Mathematical Society