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