### 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

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

# 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