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Ascent Path Rado's conjecture SNR Reduced Power full tree Ramsey theory over partitions coloring number Whitehead Problem Uniformization Countryman line Generalized Clubs Successor of Regular Cardinal very good scale Uniformly homogeneous Nonspecial tree b-scale Singular cardinals combinatorics Sierpinski's onto mapping principle Aronszajn tree incompactness Foundations Vanishing levels Ineffable cardinal Fat stationary set Chang's conjecture Diamond for trees projective Boolean algebra Hedetniemi's conjecture Subtle tree property stationary hitting Almost countably chromatic C-sequence weak diamond Successor of Singular Cardinal Prevalent singular cardinals Small forcing Square-Brackets Partition Relations Analytic sets polarized partition relation Sigma-Prikry Diamond Partition Relations PFA(S)[S] indecomposable ultrafilter Well-behaved magma ZFC construction Distributive tree Weakly compact cardinal Poset Knaster and friends Singular cofinality Open Access Chromatic number sap Iterated forcing Constructible Universe Luzin set positive partition relation free Boolean algebra Forcing Axioms strongly bounded groups Postprocessing function weak Kurepa tree Commutative cancellative semigroups Mandelbrot set Selective Ultrafilter Diamond-sharp Dowker space Club Guessing Filter reflection Erdos Cardinal Singular Density Rock n' Roll Martin's Axiom Subnormal ideal square PFA Strongly compact cardinal Minimal Walks Antichain Cohen real Cardinal function Forcing Strong coloring nonmeager set Cardinal Invariants Lipschitz reduction xbox Fast club Rainbow sets Subtle cardinal Sakurai's Bell inequality Subadditive Local Club Condensation. Generalized descriptive set theory Shelah's Strong Hypothesis free Souslin tree reflection principles Amenable C-sequence Respecting tree Non-saturation Axiom R Jonsson cardinal Dushnik-Miller middle diamond countably metacompact stationary reflection super-Souslin tree Almost-disjoint family Prikry-type forcing Strongly Luzin set Precaliber square principles Was Ulam right specializable Souslin tree club_AD approachability ideal Uniformly coherent Hindman's Theorem Erdos-Hajnal graphs transformations Parameterized proxy principle Hereditarily Lindelöf space P-Ideal Dichotomy Knaster Slim tree Ulam matrix GMA Almost Souslin regressive Souslin tree Ostaszewski square unbounded function diamond star weak square stick Kurepa Hypothesis S-Space Greatly Mahlo Coherent tree L-space Closed coloring Universal Sequences Intersection model O-space Large Cardinals HOD Absoluteness Microscopic Approach Fodor-type reflection higher Baire space OCA Souslin Tree tensor product graph 54G20 ccc AIM forcing Reflecting stationary set Commutative projection system
Tag Archives: Open Access
Complicated colorings
Abstract. If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^\lambda_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. (Recall that $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ asserts the existence of a coloring $d:[\lambda]^2\rightarrow\lambda$ such that for any family $\mathcal A\subseteq[\lambda]^{<\kappa}$ of size $\lambda$, consisting of pairwise … Continue reading
Posted in Partition Relations, Publications
Tagged Minimal Walks, Open Access, Square-Brackets Partition Relations
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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
