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