### 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

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

# 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