### 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

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

# 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