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reflection principles P-Ideal Dichotomy Diamond for trees Constructible Universe indecomposable filter Shelah's Strong Hypothesis Slim tree Interval topology on trees specializable Souslin tree Local Club Condensation. Small forcing Erdos-Hajnal graphs Knaster and friends Antichain Prikry-type forcing free Souslin tree Filter reflection weak square Minimal Walks Almost Souslin Sierpinski's onto mapping principle SNR square Axiom R Souslin Tree Microscopic Approach Hereditarily Lindelöf space Uniformly coherent Singular cardinals combinatorics super-Souslin tree Ostaszewski square projective Boolean algebra Forcing Large Cardinals OCA Forcing with side conditions stationary reflection PFA Open Access Foundations Countryman line Martin's Axiom diamond star Subadditive polarized partition relation Weakly compact cardinal Chang's conjecture Subnormal ideal middle diamond Successor of Regular Cardinal strongly bounded groups Knaster Ulam matrix weak diamond b-scale Dowker space Diamond Ascending path higher Baire space Entangled linear order countably metacompact free Boolean algebra Greatly Mahlo Uniformization Jonsson cardinal Universal Sequences 54G20 Diamond-sharp Prevalent singular cardinals square principles Aronszajn tree Chromatic number coloring number Whitehead Problem xbox positive partition relation Luzin set Dushnik-Miller Rainbow sets S-Space Commutative projection system Fast club stationary hitting Selective Ultrafilter Non-saturation Generalized Clubs Fat stationary set Cohen real weak Kurepa tree Iterated forcing Successor of Singular Cardinal Singular cofinality Kurepa Hypothesis Rock n' Roll Cardinal function sap Amenable C-sequence unbounded function Sigma-Prikry Postprocessing function incompactness Generalized descriptive set theory very good scale Nonspecial tree PFA(S)[S] Commutative cancellative semigroups Monotonically far Uniformly homogeneous L-space Erdos Cardinal AIM forcing Intersection model Well-behaved magma Respecting tree Ascent Path Partition Relations Precaliber regressive Souslin tree Hedetniemi's conjecture Reduced Power Forcing Axioms Sakurai's Bell inequality Cardinal Invariants approachability ideal Club Guessing Subtle cardinal Reflecting stationary set Ramsey theory over partitions nonmeager set transformations Distributive tree ZFC construction Subtle tree property Poset club_AD Almost countably chromatic Strongly compact cardinal Fodor-type reflection Hindman's Theorem Almost-disjoint family Lipschitz reduction Strong coloring Coherent tree Mandelbrot set ccc stick tensor product graph Vanishing levels Closed coloring Strongly Luzin set Analytic sets C-sequence Square-Brackets Partition Relations Ineffable cardinal O-space Was Ulam right? HOD Absoluteness Rado's conjecture full tree GMA Partition relations for trees Singular Density perfectly normal Parameterized proxy principle
Tag Archives: 03E05
Ketonen’s question and other cardinal sins
Joint work with Zhixing You and Jiachen Yuan. Abstract. Answering a question of Ketonen from the late 1970’s, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing … Continue reading
Fake Reflection
Joint work with Gabriel Fernandes and Miguel Moreno. Abstract. We introduce a generalization of stationary set reflection which we call filter reflection, and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We … 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
A remark on Schimmerling’s question
Joint work with Ari Meir Brodsky. Abstract. Schimmerling asked whether $\square^*_\lambda$ together with GCH entails the existence of a $\lambda^+$-Souslin tree, for a singular cardinal $\lambda$. Here, we provide an affirmative answer under the additional assumption that there exists a … Continue reading
Weak square and stationary reflection
Joint work with Gunter Fuchs. Abstract. It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, stationary reflection, weak square
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A forcing axiom deciding the generalized Souslin Hypothesis
Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
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Higher Souslin trees and the GCH, revisited
Abstract. It is proved that for every uncountable cardinal $\lambda$, GCH+$\square(\lambda^+)$ entails the existence of a $\text{cf}(\lambda)$-complete $\lambda^+$-Souslin tree. In particular, if GCH holds and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in Godel’s constructible universe, improving … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, Open Access, regressive Souslin tree, Souslin Tree, square, Weakly compact cardinal, xbox
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A microscopic approach to Souslin-tree constructions. Part I
Joint work with Ari Meir Brodsky. Abstract. We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E65, 05C05, Coherent tree, Diamond, Microscopic Approach, Parameterized proxy principle, Slim tree, Souslin Tree, square, xbox
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Square with built-in diamond-plus
Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, Respecting tree, square, xbox
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