### 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

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

# 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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