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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# Tag Archives: OCA

## 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