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Antichain positive partition relation Hindman's Theorem Martin's Axiom C-sequence AIM forcing Analytic sets Luzin set Precaliber OCA O-space Cardinal function Ascent Path Reflecting stationary set coloring number weak square Shelah's Strong Hypothesis polarized partition relation Ramsey theory over partitions Almost Souslin specializable Souslin tree Kurepa Hypothesis Souslin Tree Iterated forcing Whitehead Problem approachability ideal stationary reflection Was Ulam right diamond star Lipschitz reduction b-scale Diamond for trees Closed coloring Diamond Fodor-type reflection Knaster Subnormal ideal Uniformly coherent middle diamond Subtle tree property ZFC construction Commutative cancellative semigroups PFA stationary hitting Large Cardinals unbounded function P-Ideal Dichotomy Coherent tree tensor product graph incompactness very good scale Nonspecial tree Microscopic Approach Chang's conjecture projective Boolean algebra Well-behaved magma Club Guessing Subadditive Subtle cardinal S-Space 54G20 Singular cardinals combinatorics Partition Relations Forcing Non-saturation indecomposable ultrafilter Ineffable cardinal Rainbow sets Cohen real Strong coloring Chromatic number Open Access Small forcing Filter reflection super-Souslin tree transformations Greatly Mahlo Uniformization Hedetniemi's conjecture HOD Almost-disjoint family reflection principles full tree Singular cofinality Generalized Clubs Strongly Luzin set Almost countably chromatic GMA Ulam matrix regressive Souslin tree Minimal Walks Postprocessing function SNR Diamond-sharp Rock n' Roll stick Uniformly homogeneous Absoluteness Rado's conjecture free Souslin tree ccc PFA(S)[S] Dushnik-Miller Forcing Axioms Fast club Selective Ultrafilter free Boolean algebra Sierpinski's onto mapping principle Dowker space Sakurai's Bell inequality Successor of Singular Cardinal Foundations xbox Jonsson cardinal Slim tree Constructible Universe countably metacompact Erdos Cardinal Hereditarily Lindelöf space Generalized descriptive set theory higher Baire space square Square-Brackets Partition Relations Parameterized proxy principle Successor of Regular Cardinal Cardinal Invariants weak diamond Knaster and friends nonmeager set square principles sap Poset Vanishing levels Amenable C-sequence Ostaszewski square Erdos-Hajnal graphs Prevalent singular cardinals Weakly compact cardinal strongly bounded groups L-space Prikry-type forcing Sigma-Prikry Universal Sequences Singular Density Reduced Power Distributive tree Local Club Condensation. Axiom R Aronszajn tree Fat stationary set club_AD Mandelbrot set
Category Archives: Compactness
Squares, ultrafilters and forcing axioms
Joint work with Chris Lambie-Hanson and Jing Zhang. Abstract. We study the interplay of the three families of combinatorial objects or principles. Specifically, we show the following. Strong forcing axioms, in general incompatible with the existence of indexed squares, can … Continue reading
Posted in Compactness, Preprints
Tagged Forcing Axioms, indecomposable ultrafilter, Subadditive, unbounded function
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Sigma-Prikry forcing III: Down to Aleph_omega
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_\omega$ together with the reflection of all stationary subsets of $\aleph_{\omega+1}$. This shows that two classical results of … Continue reading
Sigma-Prikry forcing II: Iteration Scheme
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. In Part I of this series, we introduced a class of notions of forcing which we call $\Sigma$-Prikry, and showed that many of the known Prikry-type notions of forcing that centers … Continue reading
Knaster and friends II: The C-sequence number
Joint work with Chris Lambie-Hanson. Abstract. Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable … Continue reading
Sigma-Prikry forcing I: The Axioms
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality … Continue reading
The eightfold way
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading
Reflection on the coloring and chromatic numbers
Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Chang's conjecture, Chromatic number, coloring number, Fodor-type reflection, incompactness, Iterated forcing, Parameterized proxy principle, Postprocessing function, Rado's conjecture, square, stationary reflection
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Chromatic numbers of graphs – large gaps
Abstract. We say that a graph $G$ is $(\aleph_0,\kappa)$-chromatic if $\text{Chr}(G)=\kappa$, while $\text{Chr}(G’)\le\aleph_0$ for any subgraph $G’$ of $G$ of size $<|G|$. The main result of this paper reads as follows. If $\square_\lambda+\text{CH}_\lambda$ holds for a given uncountable cardinal $\lambda$, … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Almost countably chromatic, Chromatic number, incompactness, Ostaszewski square
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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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Openly generated Boolean algebras and the Fodor-type reflection principle
Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading