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