### 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

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

# 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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