Nonhomogeneous analytic families of trees
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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
- MR Author ID: 633758
- Email: James.Hirschorn@quantitative-technologies.com
- 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.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 3001-3018
- MSC (2010): Primary 03E15; Secondary 03E40, 05D40, 28A12
- DOI: https://doi.org/10.1090/tran/7517
- MathSciNet review: 3988600