### 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
- Genearlizations 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

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

# 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
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## 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
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