### 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

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

# 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