### 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
- Genearlizations 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

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

# Tag Archives: Uniformization

## Genearlizations 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