Tag Archives: Souslin Tree

Prikry forcing may add a Souslin tree

A celebrated theorem of Shelah states that adding a Cohen real introduces a Souslin tree. Are there any other examples of notions of forcing that add a κ-Souslin tree? and why is this of interest? My motivation comes from a … Continue reading

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Higher Souslin trees and the GCH, revisited

Abstract.  It is proved that for every uncountable cardinal λ, GCH+◻(λ+) entails the existence of a cf(λ)-complete λ+-Souslin tree. In particular, if GCH holds and there are no 2-Souslin trees, then 2 is weakly compact in Godel’s constructible universe, improving … Continue reading

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Prolific Souslin trees

In a paper from 1971, Erdos and Hajnal asked whether (assuming CH) every coloring witnessing 1[1]32 has a rainbow triangle. The negative solution was given in a 1975 paper by Shelah, and the proof and relevant definitions may be found … Continue reading

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A microscopic approach to Souslin-tree constructions. Part I

Joint work with Ari Meir Brodsky. Abstract.  We propose a parameterized proxy principle from which κ-Souslin trees with various additional features can be constructed, regardless of the identity of κ. We then introduce the microscopic approach, which is a simple … Continue reading

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P.O.I. Workshop in pure and descriptive set theory, September 2015

I gave an invited talk at the P.O.I Workshop in pure and descriptive set theory, Torino, September 26, 2015.  Title: 3-trees. Abstract: We inspect the constructions of four quite different 3-Souslin trees.

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Reduced powers of Souslin trees

Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a κ-Souslin tree T and its reduced powers Tθ/U. Previous works addressed this problem from the viewpoint of a single power θ, whereas here, tools are developed … Continue reading

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Forcing and its Applications Retrospective Workshop, April 2015

I gave an invited talk at Forcing and its Applications Retrospective Workshop, Toronto, April 1st, 2015.  Title: A microscopic approach to Souslin trees constructions Abstract: We present an approach to construct κ-Souslin trees that is insensitive to the identity of … Continue reading

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Forcing with a Souslin tree makes p=ω1

I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a … Continue reading

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Forcing with a Souslin tree makes p=ω1

I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a … Continue reading

Posted in Blog, Expository | Tagged | 2 Comments

The P-Ideal Dichotomy and the Souslin Hypothesis

John Krueger is visiting Toronto these days, and in a conversation today, we asked ourselves how do one prove the Abraham-Todorcevic theorem that PID implies SH. Namely, that the next statement implies that there are no Souslin trees: Definition. The … Continue reading

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