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Hindman's Theorem Subnormal ideal Kurepa Hypothesis Analytic sets AIM forcing Uniformly coherent stationary hitting Jonsson cardinal higher Baire space Club Guessing Poset Forcing Axioms Diamond-sharp Minimal Walks b-scale Parameterized proxy principle Cohen real Selective Ultrafilter transformations Respecting tree Closed coloring weak square Generalized descriptive set theory Diamond Strongly compact cardinal Singular Density Strong coloring regressive Souslin tree Local Club Condensation. Large Cardinals Almost Souslin C-sequence Singular cardinals combinatorics Distributive tree Cardinal function Fast club Subadditive middle diamond O-space Was Ulam right Small forcing weak Kurepa tree free Souslin tree club_AD Antichain ZFC construction Weakly compact cardinal square principles Hedetniemi's conjecture Martin's Axiom Ineffable cardinal stick unbounded function Postprocessing function SNR Uniformly homogeneous full tree polarized partition relation P-Ideal Dichotomy Cardinal Invariants Countryman line Subtle cardinal strongly bounded groups Rado's conjecture Knaster Ulam matrix Amenable C-sequence Intersection model Iterated forcing Reflecting stationary set Knaster and friends Almost-disjoint family Precaliber ccc positive partition relation PFA Prevalent singular cardinals Shelah's Strong Hypothesis Rainbow sets Dowker space Whitehead Problem Commutative projection system Reduced Power indecomposable ultrafilter Forcing Diamond for trees Coherent tree Dushnik-Miller Constructible Universe Aronszajn tree HOD Square-Brackets Partition Relations diamond star xbox Well-behaved magma Microscopic Approach reflection principles very good scale Ostaszewski square Almost countably chromatic L-space tensor product graph Strongly Luzin set Erdos-Hajnal graphs Filter reflection Absoluteness Nonspecial tree projective Boolean algebra incompactness Generalized Clubs square nonmeager set Sierpinski's onto mapping principle Vanishing levels Singular cofinality 54G20 S-Space Hereditarily Lindelöf space specializable Souslin tree Fodor-type reflection Rock n' Roll Universal Sequences Successor of Regular Cardinal Slim tree free Boolean algebra Successor of Singular Cardinal Partition Relations Uniformization Greatly Mahlo Mandelbrot set Luzin set Open Access GMA Chromatic number Lipschitz reduction Commutative cancellative semigroups Ramsey theory over partitions Sakurai's Bell inequality coloring number countably metacompact Erdos Cardinal Prikry-type forcing weak diamond PFA(S)[S] Souslin Tree stationary reflection OCA super-Souslin tree sap Sigma-Prikry Fat stationary set Non-saturation Subtle tree property Foundations Ascent Path Axiom R Chang's conjecture approachability ideal
Category Archives: Compactness
Ketonen’s question and other cardinal sins
Joint work with Zhixing You and Jiachen Yuan. Abstract. Intersection models of generic extensions obtained from a commutative projection systems of notions of forcing has recently regained interest, especially in the study of descriptive set theory. Here, we show that … Continue reading
Posted in Compactness, Preprints
Tagged 03E05, 03E35, 03E55, Ascent Path, C-sequence, Commutative projection system, indecomposable ultrafilter, Intersection model, Strongly compact cardinal
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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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