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