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AIM forcing Amenable C-sequence Cohen real L-space Filter reflection Well-behaved magma Jonsson cardinal S-Space Weakly compact cardinal stationary reflection indecomposable filter ccc HOD transformations Cardinal function Uniformization Coherent tree Martin's Axiom Kurepa Hypothesis Square-Brackets Partition Relations Local Club Condensation. Monotonically far Knaster and friends Successor of Singular Cardinal Uniformly homogeneous Open Access full tree Interval topology on trees Small forcing Strongly compact cardinal Singular Density Chromatic number Respecting tree Partition Relations Souslin Tree Generalized Clubs positive partition relation Countryman line Commutative projection system weak square Distributive tree Forcing Axioms Almost-disjoint family Diamond stationary hitting diamond star Ascending path Rainbow sets Fodor-type reflection PFA projective Boolean algebra Constructible Universe perfectly normal Uniformly coherent middle diamond Analytic sets polarized partition relation Subadditive Nonspecial tree Subtle cardinal GMA Rock n' Roll super-Souslin tree Club Guessing Partition relations for trees weak Kurepa tree Poset Subtle tree property Hereditarily Lindelöf space Shelah's Strong Hypothesis Hedetniemi's conjecture Rado's conjecture Forcing with side conditions countably metacompact Parameterized proxy principle Strongly Luzin set Fast club Commutative cancellative semigroups Ineffable cardinal Fat stationary set Large Cardinals Iterated forcing Subnormal ideal Ascent Path Mandelbrot set Erdos Cardinal Intersection model Whitehead Problem SNR Non-saturation ZFC construction Entangled linear order coloring number Ulam matrix Singular cofinality Slim tree Sigma-Prikry Precaliber free Souslin tree free Boolean algebra Luzin set P-Ideal Dichotomy square unbounded function Successor of Regular Cardinal Dushnik-Miller Erdos-Hajnal graphs Universal Sequences Strong coloring Minimal Walks Knaster Almost Souslin Absoluteness square principles Lipschitz reduction 54G20 b-scale Sierpinski's onto mapping principle Ostaszewski square higher Baire space Dowker space Reflecting stationary set Antichain nonmeager set incompactness Postprocessing function club_AD specializable Souslin tree Diamond for trees strongly bounded groups Reduced Power Prevalent singular cardinals O-space Closed coloring reflection principles Vanishing levels Generalized descriptive set theory Foundations xbox regressive Souslin tree very good scale Prikry-type forcing Singular cardinals combinatorics Axiom R PFA(S)[S] C-sequence Sakurai's Bell inequality Ramsey theory over partitions Hindman's Theorem Greatly Mahlo Diamond-sharp Microscopic Approach approachability ideal Aronszajn tree Cardinal Invariants Forcing sap tensor product graph Chang's conjecture Almost countably chromatic weak diamond stick OCA Was Ulam right? Selective Ultrafilter
Category Archives: Publications
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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Distributive Aronszajn trees
Joint work with Ari Meir Brodsky. Abstract. Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved that the same conclusion remains … 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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Strong failures of higher analogs of Hindman’s Theorem
Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading
Posted in Groups, Partition Relations, Publications
Tagged 03E02, 03E35, 03E75, 05A17, 05D10, 11P99, 20M14, Chang's conjecture, Commutative cancellative semigroups, Entangled linear order, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal, ZFC construction
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More notions of forcing add a Souslin tree
Joint work with Ari Meir Brodsky. Abstract. An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing … Continue reading
Ordinal definable subsets of singular cardinals
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such … Continue reading
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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