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square Uniformly homogeneous Reduced Power nonmeager set countably metacompact b-scale Strong coloring Monotonically far Singular cardinals combinatorics Small forcing Generalized descriptive set theory Local Club Condensation. Successor of Singular Cardinal PFA(S)[S] Countryman line Hindman's Theorem Postprocessing function Rock n' Roll Subadditive weak square Lipschitz reduction Almost Souslin S-Space strongly bounded groups regressive Souslin tree P-Ideal Dichotomy Coherent tree Kurepa Hypothesis stick Fodor-type reflection sap specializable Souslin tree Intersection model Open Access Diamond-sharp Weakly compact cardinal Whitehead Problem Ramsey theory over partitions Microscopic Approach projective Boolean algebra Rainbow sets OCA Minimal Walks AIM forcing polarized partition relation HOD Jonsson cardinal super-Souslin tree Non-saturation perfectly normal C-sequence Commutative projection system higher Baire space xbox Erdos-Hajnal graphs diamond star Sierpinski's onto mapping principle indecomposable filter incompactness Cardinal Invariants Prikry-type forcing Hereditarily Lindelöf space Sigma-Prikry PFA Ostaszewski square Erdos Cardinal Generalized Clubs Commutative cancellative semigroups Poset Fast club Absoluteness very good scale coloring number Nonspecial tree Diamond Ascending path Knaster and friends unbounded function Iterated forcing Partition Relations Shelah's Strong Hypothesis Singular Density Axiom R Filter reflection Rado's conjecture Souslin Tree Strongly compact cardinal ccc Diamond for trees Foundations Knaster approachability ideal Forcing with side conditions Hedetniemi's conjecture Precaliber weak Kurepa tree Reflecting stationary set Subtle tree property Singular cofinality weak diamond Dowker space Parameterized proxy principle Slim tree Mandelbrot set Forcing Axioms transformations middle diamond Uniformization stationary reflection Universal Sequences GMA Uniformly coherent free Boolean algebra Interval topology on trees Amenable C-sequence Chang's conjecture Ineffable cardinal Was Ulam right? Subnormal ideal O-space Square-Brackets Partition Relations Chromatic number Strongly Luzin set Luzin set tensor product graph Almost countably chromatic club_AD Partition relations for trees L-space Almost-disjoint family ZFC construction Cardinal function Respecting tree Selective Ultrafilter Prevalent singular cardinals reflection principles Ascent Path Large Cardinals Distributive tree Greatly Mahlo Fat stationary set Sakurai's Bell inequality Vanishing levels Constructible Universe square principles Closed coloring Forcing Aronszajn tree Club Guessing full tree stationary hitting 54G20 Well-behaved magma Cohen real positive partition relation Martin's Axiom Successor of Regular Cardinal Dushnik-Miller SNR free Souslin tree Ulam matrix Subtle cardinal Antichain Analytic sets Entangled linear order
Category Archives: Publications
Partitioning a reflecting stationary set
Joint work with Maxwell Levine. Abstract. We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer … Continue reading
Souslin trees at successors of regular cardinals
Abstract. We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author. Downloads: Citation … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged Parameterized proxy principle, Souslin Tree
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Knaster and friends I: Closed colorings and precalibers
Joint work with Chris Lambie-Hanson. Abstract. The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $\kappa$-cc posets whose squares … 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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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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