### 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

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

# 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