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### 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

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

## The uniformization property for $\aleph_2$

## 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