### Archives

### 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

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

# Tag Archives: Uniformization

## 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

## The uniformization property for $\aleph_2$

Given a subset of a regular uncountable cardinal $S\subseteq\kappa$, $UP_S$ (read: “the uniformization property holds for $S$”) asserts that for every sequence $\overrightarrow f=\langle f_\alpha\mid \alpha\in S\rangle$ satisfying for all $\alpha\in S$: $f_\alpha$ is a 2-valued function; $\text{dom}(f_\alpha)$ is a … 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

## Jensen’s diamond principle and its relatives

This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … Continue reading

## On guessing generalized clubs at the successors of regulars

Abstract: Konig, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of an higher Souslin tree from the strong guessing principle. Complementary to the author’s work on the validity of diamond and non-saturation … Continue reading