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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

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

# Tag Archives: ccc

## Generalizations of Martin’s Axiom and the well-met condition

Recall that Martin’s Axiom asserts that for every partial order $\mathbb P$ satisfying c.c.c., and for any family $\mathcal D$ of $<2^{\aleph_0}$ many dense subsets of $\mathbb P$, there exists a directed subset $G$ of $\mathbb P$ such that $G\cap … Continue reading

## Chain conditions of products, and weakly compact cardinals

Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading

Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
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## 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

## c.c.c. forcing without combinatorics

In this post, we shall discuss a short paper by Alan Mekler from 1984, concerning a non-combinatorial verification of the c.c.c. property for forcing notions. Recall that a notion of forcing $\mathbb P$ is said to satisfy the c.c.c. iff … Continue reading