### 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

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

# Tag Archives: b-scale

## 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

## 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