### 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

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

# 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