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Partition relations for trees strongly bounded groups indecomposable filter middle diamond Square-Brackets Partition Relations Greatly Mahlo Postprocessing function Closed coloring HOD Strongly compact cardinal Uniformly coherent Singular Density free Souslin tree Universal Sequences Chromatic number SNR regressive Souslin tree Whitehead Problem Singular cardinals combinatorics reflection principles Club Guessing approachability ideal Hindman's Theorem Fat stationary set Filter reflection Sakurai's Bell inequality Chang's conjecture Almost-disjoint family Hereditarily Lindelöf space club_AD Non-saturation Subnormal ideal O-space Large Cardinals weak diamond Prevalent singular cardinals Hedetniemi's conjecture coloring number specializable Souslin tree PFA(S)[S] Parameterized proxy principle sap Ascent Path Distributive tree Sigma-Prikry GMA Singular cofinality Rock n' Roll positive partition relation Amenable C-sequence stationary reflection Analytic sets Shelah's Strong Hypothesis Ramsey theory over partitions weak Kurepa tree Small forcing Interval topology on trees Foundations ZFC construction Fodor-type reflection countably metacompact Knaster and friends Sierpinski's onto mapping principle Respecting tree Dowker space Subadditive Cardinal function Vanishing levels Constructible Universe Precaliber Diamond Partition Relations Open Access Ostaszewski square Uniformization Kurepa Hypothesis Aronszajn tree Countryman line nonmeager set Minimal Walks Commutative cancellative semigroups Martin's Axiom weak square Reflecting stationary set Nonspecial tree unbounded function Generalized descriptive set theory Commutative projection system Uniformly homogeneous Forcing Axioms Selective Ultrafilter square Mandelbrot set Slim tree Ulam matrix Almost countably chromatic projective Boolean algebra Strong coloring tensor product graph Weakly compact cardinal AIM forcing incompactness Generalized Clubs Rainbow sets Lipschitz reduction Luzin set Poset Ascending path Subtle tree property Diamond for trees super-Souslin tree S-Space polarized partition relation Well-behaved magma b-scale Absoluteness Intersection model ccc Monotonically far Souslin Tree Rado's conjecture Coherent tree Strongly Luzin set Cardinal Invariants transformations PFA stationary hitting Reduced Power Successor of Regular Cardinal Was Ulam right? Successor of Singular Cardinal Subtle cardinal Ineffable cardinal Almost Souslin 54G20 Cohen real Jonsson cardinal Forcing with side conditions Forcing L-space C-sequence P-Ideal Dichotomy xbox Entangled linear order diamond star Local Club Condensation. square principles Knaster Microscopic Approach Erdos-Hajnal graphs Erdos Cardinal very good scale OCA Axiom R stick Iterated forcing higher Baire space Dushnik-Miller Diamond-sharp Antichain free Boolean algebra Fast club full tree perfectly normal Prikry-type forcing
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
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Tagged ccc, Forcing Axioms, GMA, Martin's Axiom, Uniformization
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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
