### 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

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

# 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