Archives
Keywords
Cardinal Invariants Ramsey theory over partitions coloring number Slim tree Intersection model Aronszajn tree Subtle tree property weak square Nonspecial tree Hedetniemi's conjecture Filter reflection Square-Brackets Partition Relations Strongly compact cardinal Ascent Path strongly bounded groups Reflecting stationary set Commutative projection system SNR projective Boolean algebra Distributive tree Countryman line square principles Minimal Walks Uniformization ccc tensor product graph Postprocessing function Reduced Power Almost-disjoint family PFA Jonsson cardinal Cohen real Whitehead Problem Respecting tree Lipschitz reduction Baumgartner's square Microscopic Approach Generalized Clubs stationary hitting Amenable C-sequence Rado's conjecture Local Club Condensation. approachability ideal Fodor-type reflection Subtle cardinal unbounded function b-scale Forcing Axioms Knaster and friends Singular cofinality Uniformly coherent Successor of Regular Cardinal Fast club Parameterized proxy principle Singular cardinals combinatorics HOD PFA(S)[S] Ostaszewski square club_AD Forcing with side conditions super-Souslin tree Interval topology on trees Martin's Axiom regressive Souslin tree ZFC construction Open Access Strongly Luzin set Axiom R stationary reflection free Boolean algebra Precaliber P-Ideal Dichotomy reflection principles Subnormal ideal S-Space Cardinal function Rock n' Roll specializable Souslin tree Sierpinski's onto mapping principle Hindman's Theorem Uniformly homogeneous Luzin set polarized partition relation Partition Relations xbox incompactness weak Kurepa tree Selective Ultrafilter Souslin Tree nonmeager set Chromatic number Singular Density stick O-space Diamond-sharp perfectly normal Diamond Absoluteness square GMA L-space Forcing Subadditive Antichain Prevalent singular cardinals Knaster Fat stationary set Vanishing levels AIM forcing Hereditarily Lindelöf space very good scale Prikry-type forcing Closed coloring free Souslin tree Ascending path Rainbow sets Was Ulam right? diamond star middle diamond higher Baire space C-sequence countably metacompact Non-saturation Dushnik-Miller Ulam matrix Large Cardinals Erdos-Hajnal graphs Poset Almost countably chromatic Generalized descriptive set theory Universal Sequences 54G20 Sakurai's Bell inequality Well-behaved magma Entangled linear order Successor of Singular Cardinal Constructible Universe Shelah's Strong Hypothesis Diamond for trees Coherent tree Kurepa Hypothesis Erdos Cardinal transformations Greatly Mahlo Monotonically far Ineffable cardinal weak diamond indecomposable filter Mandelbrot set Chang's conjecture Dowker space Partition relations for trees Sigma-Prikry Small forcing Iterated forcing Weakly compact cardinal Foundations Almost Souslin positive partition relation OCA Analytic sets full tree Commutative cancellative semigroups sap Club Guessing Strong coloring
Tag Archives: Open Access
Aspects of singular cofinality
Abstract. We study properties of closure operators of singular cofinality, and introduce several ZFC sufficient and equivalent conditions for the existence of antichain sequences in posets of singular cofinality. We also notice that the Proper Forcing Axiom implies the Milner-Sauer … Continue reading
On topological spaces of singular density and minimal weight
Abstract: We introduce a weakening of the Generalized Continuum Hypothesis, which we will refer to as the Prevalent Singular cardinals Hypothesis (PSH), and show it implies that every topological space of density and weight $\aleph_{\omega_1}$ is not hereditarily Lindelöf. The assumption … Continue reading
A topological reflection principle equivalent to Shelah’s strong hypothesis
Abstract: We notice that Shelah’s Strong Hypothesis (SSH) is equivalent to the following reflection principle: Suppose $\mathbb X$ is an (infinite) first-countable space whose density is a regular cardinal, $\kappa$. If every separable subspace of $\mathbb X$ is of cardinality at most … Continue reading
Posted in Compactness, Publications, Topology
Tagged 03E04, 03E65, 54G15, Open Access, Shelah's Strong Hypothesis
Comments Off on A topological reflection principle equivalent to Shelah’s strong hypothesis
The failure of diamond on a reflecting stationary set
Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with … 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
On the consistency strength of the Milner-Sauer conjecture
Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The conjecture is consistent and known … Continue reading
Transforming rectangles into squares, with applications to strong colorings
Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading
