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

# 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