### 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

HOD Absoluteness Singular coﬁnality Rock n' Roll xbox free Boolean algebra Minimal Walks Foundations weak diamond weak square Microscopic Approach Parameterized proxy principle Kurepa Hypothesis Diamond Forcing Axioms Aronszajn tree Stevo Todorcevic Hedetniemi's conjecture projective Boolean algebra Axiom R Postprocessing function Large Cardinals Forcing Reduced Power Prikry-type forcing ccc Chromatic number 11P99 Mandelbrot set Uniformization polarized partition relation square principles Almost-disjoint famiy Slim tree Cohen real Shelah's Strong Hypothesis Cardinal Invariants Rado's conjecture Small forcing Chang's conjecture middle diamond 20M14 Distributive tree stationary hitting Constructible Universe Fat stationary set sap Knaster Antichain coloring number PFA tensor product graph Successor of Regular Cardinal Singular cardinals combinatorics Sakurai's Bell inequality 05A17 Nonspecial tree Uniformly coherent Hereditarily Lindelöf space approachability ideal Square-Brackets Partition Relations Dushnik-Miller Selective Ultrafilter Weakly compact cardinal P-Ideal Dichotomy Commutative cancellative semigroups Whitehead Problem Almost Souslin Coherent tree square very good scale Ostaszewski square Almost countably chromatic Prevalent singular cardinals Non-saturation Fast club S-Space Fodor-type reflection Martin's Axiom L-space Generalized Clubs diamond star Souslin Tree Club Guessing Cardinal function reflection principles Erdos-Hajnal graphs PFA(S)[S] Successor of Singular Cardinal Erdos Cardinal 05D10 Ascent Path b-scale Jonsson cardinal Universal Sequences Partition Relations Rainbow sets Singular Density stationary reflection Hindman's Theorem incompactness OCA Poset

# Tag Archives: b-scale

## 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
2 Comments