### Archives

### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Generalizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014

### Keywords

incompactness Forcing Axioms ccc diamond star Minimal Walks Absoluteness Hereditarily Lindelöf space reflection principles Club Guessing Singular coﬁnality Antichain free Souslin tree Cardinal function b-scale Stevo Todorcevic Fodor-type reflection Commutative cancellative semigroups Poset Forcing free Boolean algebra PFA tensor product graph Coherent tree Mandelbrot set Knaster Non-saturation Ostaszewski square Whitehead Problem coloring number Reduced Power Selective Ultrafilter weak square HOD stationary reflection middle diamond xbox Postprocessing function Singular Density Uniformization Uniformly coherent square principles Ascent Path Distributive tree Cardinal Invariants very good scale Square-Brackets Partition Relations polarized partition relation Parameterized proxy principle stationary hitting Fast club Almost-disjoint famiy Prevalent singular cardinals Chromatic number Successor of Singular Cardinal super-Souslin tree Aronszajn tree Singular cardinals combinatorics Nonspecial tree Microscopic Approach Almost Souslin Partition Relations Successor of Regular Cardinal Rainbow sets Shelah's Strong Hypothesis Chang's conjecture OCA Cohen real Prikry-type forcing Large Cardinals square Axiom R Foundations Erdos-Hajnal graphs Hindman's Theorem Slim tree Hedetniemi's conjecture Fat stationary set Souslin Tree Almost countably chromatic projective Boolean algebra specializable Souslin tree Erdos Cardinal Rado's conjecture S-Space Universal Sequences Kurepa Hypothesis Jonsson cardinal sap PFA(S)[S] Small forcing Generalized Clubs Constructible Universe Weakly compact cardinal Luzin set Dushnik-Miller weak diamond P-Ideal Dichotomy L-space Diamond Rock n' Roll approachability ideal Martin's Axiom Sakurai's Bell inequality

# Tag Archives: Souslin Tree

## A forcing axiom deciding the generalized Souslin Hypothesis

Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading

Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
1 Comment

## 6th European Set Theory Conference, July 2017

I gave a 3-lecture tutorial at the 6th European Set Theory Conference in Budapest, July 2017. Title: Strong colorings and their applications. Abstract. Consider the following questions. Is the product of two $\kappa$-cc partial orders again $\kappa$-cc? Does there exist … Continue reading

Posted in Invited Talks, Open Problems
Tagged b-scale, Cohen real, Luzin set, Minimal Walks, Souslin Tree, Square-Brackets Partition Relations
4 Comments

## ASL North American Meeting, March 2017

I gave a plenary talk at the 2017 ASL North American Meeting in Boise, March 2017. Talk Title: The current state of the Souslin problem. Abstract: Recall that the real line is that unique separable, dense linear ordering with no endpoints in … Continue reading

## Set Theory and its Applications in Topology, September 2016

I gave an invited talk at the Set Theory and its Applications in Topology meeting, Oaxaca, September 11-16, 2016. The talk was on the $\aleph_2$-Souslin problem. If you are interested in seeing the effect of a jet lag, the video is … Continue reading

## More notions of forcing add a Souslin tree

Joint work with Ari Meir Brodsky. Abstract. An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing … Continue reading

## 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 $\kappa$-Souslin tree? and why is this of interest? My motivation comes from a … Continue reading

## Higher Souslin trees and the GCH, revisited

Abstract. It is proved that for every uncountable cardinal $\lambda$, GCH+$\square(\lambda^+)$ entails the existence of a $\text{cf}(\lambda)$-complete $\lambda^+$-Souslin tree. In particular, if GCH holds and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in Godel’s constructible universe, improving … Continue reading

Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, Souslin Tree, square, Weakly compact cardinal, xbox
16 Comments

## Prolific Souslin trees

In a paper from 1971, Erdos and Hajnal asked whether (assuming CH) every coloring witnessing $\aleph_1\nrightarrow[\aleph_1]^2_3$ 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

Posted in Blog, Expository
Tagged Rainbow sets, Souslin Tree, Square-Brackets Partition Relations
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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 $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple … Continue reading

Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E65, 05C05, Coherent tree, Diamond, Microscopic Approach, Parameterized proxy principle, Slim tree, Souslin Tree, square, xbox
4 Comments

## 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: $\aleph_3$-trees. Abstract: We inspect the constructions of four quite different $\aleph_3$-Souslin trees.