### Archives

### Recent blog posts

- 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
- Partitioning the club guessing January 22, 2014

### Keywords

PFA(S)[S] Successor of Regular Cardinal Selective Ultrafilter b-scale projective Boolean algebra weak diamond Sakurai's Bell inequality Universal Sequences Singular coﬁnality Kurepa Hypothesis ccc middle diamond Prevalent singular cardinals coloring number Nonspecial tree polarized partition relation Uniformly coherent Diamond Large Cardinals Dushnik-Miller Stevo Todorcevic tensor product graph Almost-disjoint famiy Minimal Walks OCA Mandelbrot set Erdos Cardinal Cohen real Club Guessing Luzin set reflection principles P-Ideal Dichotomy Rock n' Roll Almost countably chromatic Square-Brackets Partition Relations Ostaszewski square Souslin Tree Hindman's Theorem Coherent tree Non-saturation square weak square free Boolean algebra Rado's conjecture L-space Microscopic Approach Rainbow sets Cardinal Invariants Distributive tree Cardinal function Chang's conjecture Small forcing Hereditarily Lindelöf space Forcing Fast club Chromatic number Erdos-Hajnal graphs Reduced Power Axiom R incompactness Aronszajn tree Prikry-type forcing Whitehead Problem Slim tree 20M14 stationary reflection Shelah's Strong Hypothesis Ascent Path Jonsson cardinal Absoluteness super-Souslin tree PFA Postprocessing function Poset 11P99 very good scale Almost Souslin Knaster Singular cardinals combinatorics sap Partition Relations Hedetniemi's conjecture HOD Fat stationary set 05A17 Commutative cancellative semigroups Forcing Axioms Successor of Singular Cardinal Singular Density square principles Weakly compact cardinal Antichain Uniformization Martin's Axiom approachability ideal Generalized Clubs Foundations Fodor-type reflection stationary hitting xbox S-Space Parameterized proxy principle Constructible Universe diamond star

# Tag Archives: b-scale

## 6th European Set Theory Conference, July 2017

I gave a 3-lectures 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

## Open coloring and the cardinal invariant $\mathfrak b$

Nik Weaver asked for a direct proof of the fact that Todorcevic’s axiom implies the failure of CH fails. Here goes. Notation. For a set $X$, we write $[X]^2$ for the set of unordered pairs $\{ \{x,x’\}\mid x,x’\in X, x\neq … Continue reading

## The S-space problem, and the cardinal invariant $\mathfrak b$

Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading

## c.c.c. vs. the Knaster property

After my previous post on Mekler’s characterization of c.c.c. notions of forcing, Sam, Mike and myself discussed the value of it . We noticed that a prevalent verification of the c.c.c. goes like this: given an uncountable set of conditions, … Continue reading

## Dushnik-Miller for regular cardinals (part 2)

In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading

Posted in Blog, Expository
Tagged b-scale, Dushnik-Miller, Partition Relations, Square-Brackets Partition Relations
5 Comments

## Infinite Combinatorial Topology

Back in 2005, as a master student, I attended a course by Boaz Tsaban, entitled “Infinite Combinatorial Topology”. A friend and I decided to produce lecture notes, but in a somewhat loose sense, that is: we sometimes omit material given … Continue reading

Posted in Notes
Tagged b-scale, Cardinal function, Cardinal Invariants, Hereditarily Lindelöf space
8 Comments