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strongly bounded groups Lipschitz reduction Rainbow sets b-scale Distributive tree Local Club Condensation. Diamond-sharp Intersection model Interval topology on trees Strongly Luzin set Open Access Forcing Axioms Singular Density Monotonically far specializable Souslin tree Respecting tree Partition relations for trees Minimal Walks Uniformly homogeneous ZFC construction Sierpinski's onto mapping principle free Boolean algebra Ascending path Chang's conjecture GMA Parameterized proxy principle ccc stick Nonspecial tree OCA regressive Souslin tree Singular cardinals combinatorics polarized partition relation Square-Brackets Partition Relations xbox Absoluteness incompactness Dowker space Partition Relations Dushnik-Miller Diamond Closed coloring countably metacompact Subtle cardinal approachability ideal Knaster Strong coloring perfectly normal L-space very good scale Uniformization Generalized descriptive set theory Whitehead Problem Hedetniemi's conjecture Generalized Clubs Ostaszewski square Martin's Axiom Countryman line Antichain Almost countably chromatic super-Souslin tree Sakurai's Bell inequality Sigma-Prikry square transformations Commutative projection system Analytic sets higher Baire space Subnormal ideal Singular cofinality full tree weak square Fat stationary set S-Space Mandelbrot set Slim tree projective Boolean algebra Well-behaved magma diamond star Large Cardinals Iterated forcing AIM forcing Diamond for trees Hereditarily Lindelöf space Cardinal Invariants Cohen real Uniformly coherent tensor product graph Small forcing C-sequence Selective Ultrafilter Erdos Cardinal Kurepa Hypothesis Souslin Tree Almost-disjoint family Cardinal function Non-saturation Postprocessing function Filter reflection Knaster and friends Poset free Souslin tree middle diamond Weakly compact cardinal Chromatic number club_AD HOD Ulam matrix Forcing Fodor-type reflection Ineffable cardinal Amenable C-sequence Axiom R Forcing with side conditions square principles Foundations stationary hitting Was Ulam right? sap Reflecting stationary set Luzin set unbounded function 54G20 P-Ideal Dichotomy Rock n' Roll Ascent Path Microscopic Approach PFA Fast club Ramsey theory over partitions Greatly Mahlo Hindman's Theorem Coherent tree Successor of Singular Cardinal Aronszajn tree Precaliber weak diamond Strongly compact cardinal PFA(S)[S] indecomposable filter O-space Club Guessing Commutative cancellative semigroups Prevalent singular cardinals Subtle tree property Entangled linear order Constructible Universe Rado's conjecture Shelah's Strong Hypothesis Jonsson cardinal Successor of Regular Cardinal weak Kurepa tree Vanishing levels SNR Prikry-type forcing Almost Souslin coloring number Subadditive stationary reflection reflection principles Universal Sequences Erdos-Hajnal graphs positive partition relation nonmeager set Reduced Power
Category Archives: Souslin Hypothesis
Proxy principles in combinatorial set theory
Joint work with Ari Meir Brodsky and Shira Yadai. Abstract. The parameterized proxy principles were introduced by Brodsky and Rinot in a 2017 paper as new foundations for the construction of $\kappa$-Souslin trees in a uniform way that does not … Continue reading
The vanishing levels of a tree
Joint work with Shira Yadai and Zhixing You. Abstract. We initiate the study of the spectrum of sets that can be realized as the vanishing levels $V(\mathbf T)$ of a normal $\kappa$-tree $\mathbf T$. This is an invariant in the … Continue reading
Posted in Preprints, Souslin Hypothesis
Tagged Almost-disjoint family, Ascent Path, C-sequence, Coherent tree, Dowker space, Open Access, Parameterized proxy principle, regressive Souslin tree, Respecting tree, Subtle tree property, Uniformly homogeneous, Vanishing levels, weak Kurepa tree
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Full Souslin trees at small cardinals
Joint work with Shira Yadai and Zhixing You. Abstract. A $\kappa$-tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $\kappa$-Souslin tree may consistently exist. Shelah gave an affirmative … Continue reading
On the ideal J[kappa]
Abstract. Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[\kappa]$, from which we confirm that $\kappa$-Souslin trees exist in various models of interest. As a corollary we … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged Cardinal Invariants, Cohen real, nonmeager set
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A microscopic approach to Souslin-tree constructions. Part II
Joint work with Ari Meir Brodsky. Abstract. In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known $\diamondsuit$-based constructions of Souslin trees with various additional properties may be rendered as applications of … 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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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
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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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
