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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

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

# 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