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Filter reflection Hereditarily Lindelöf space weak square Singular Density strongly bounded groups Generalized Clubs Almost-disjoint family Intersection model Almost countably chromatic weak Kurepa tree Was Ulam right? Subtle cardinal SNR stick Subtle tree property Kurepa Hypothesis Subnormal ideal higher Baire space Whitehead Problem polarized partition relation Fat stationary set Interval topology on trees Fodor-type reflection approachability ideal Hindman's Theorem Knaster Hedetniemi's conjecture Parameterized proxy principle Singular cofinality coloring number Chang's conjecture Prevalent singular cardinals Sakurai's Bell inequality Erdos-Hajnal graphs 54G20 reflection principles Dushnik-Miller middle diamond Knaster and friends Monotonically far Almost Souslin Universal Sequences xbox Strong coloring square C-sequence Well-behaved magma Analytic sets Forcing Countryman line Fast club Uniformly coherent Generalized descriptive set theory countably metacompact Subadditive Diamond sap Aronszajn tree Shelah's Strong Hypothesis square principles Diamond for trees Square-Brackets Partition Relations unbounded function Distributive tree PFA(S)[S] Precaliber HOD projective Boolean algebra Sigma-Prikry Partition Relations full tree Reflecting stationary set very good scale Lipschitz reduction Non-saturation Rock n' Roll Iterated forcing Weakly compact cardinal stationary hitting Ulam matrix positive partition relation Constructible Universe Ramsey theory over partitions Successor of Regular Cardinal Amenable C-sequence Microscopic Approach Respecting tree Slim tree Strongly compact cardinal Antichain Luzin set Erdos Cardinal Forcing with side conditions diamond star Mandelbrot set Axiom R club_AD Entangled linear order Partition relations for trees Vanishing levels ZFC construction Closed coloring Successor of Singular Cardinal nonmeager set S-Space ccc Local Club Condensation. free Boolean algebra Nonspecial tree free Souslin tree Reduced Power Ascending path Chromatic number Postprocessing function tensor product graph Jonsson cardinal Uniformization Ostaszewski square Cardinal function Dowker space Singular cardinals combinatorics O-space Rainbow sets Open Access OCA perfectly normal Commutative projection system L-space b-scale Strongly Luzin set P-Ideal Dichotomy GMA Selective Ultrafilter Cohen real Small forcing Ineffable cardinal indecomposable filter specializable Souslin tree Rado's conjecture stationary reflection Large Cardinals transformations Prikry-type forcing weak diamond incompactness Foundations Ascent Path Souslin Tree Uniformly homogeneous Forcing Axioms super-Souslin tree AIM forcing Greatly Mahlo Absoluteness Sierpinski's onto mapping principle Commutative cancellative semigroups regressive Souslin tree Cardinal Invariants Diamond-sharp Club Guessing PFA Poset Martin's Axiom Minimal Walks Coherent tree
Tag Archives: xbox
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
Strongest transformations
Joint work with Jing Zhang. Abstract. We continue our study of maps transforming high-dimensional complicated objects into squares of stationary sets. Previously, we proved that many such transformations exist in ZFC, and here we address the consistency of the strongest … Continue reading
Posted in Partition Relations, Publications
Tagged Diamond, Minimal Walks, square, Square-Brackets Partition Relations, stick, transformations, xbox
2 Comments
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
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
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
16 Comments
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
5 Comments
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
1 Comment
Square principles
Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle … Continue reading
