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### 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

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

# 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