### Archives

### 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

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

# 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
2 Comments

## 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