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Slim tree polarized partition relation Was Ulam right xbox Knaster and friends Cardinal Invariants PFA Open Access Coherent tree Diamond-sharp very good scale unbounded function diamond star coloring number Prevalent singular cardinals strongly bounded groups Successor of Regular Cardinal projective Boolean algebra stick indecomposable ultrafilter tensor product graph Postprocessing function Forcing full tree Universal Sequences ZFC construction Dowker space Iterated forcing Uniformization Rado's conjecture Whitehead Problem Souslin Tree Constructible Universe Ramsey theory over partitions Uniformly coherent Generalized Clubs Erdos-Hajnal graphs Kurepa Hypothesis Hindman's Theorem Small forcing Club Guessing Almost Souslin C-sequence stationary reflection Distributive tree AIM forcing Uniformly homogeneous Axiom R Erdos Cardinal Lipschitz reduction Cardinal function Singular cardinals combinatorics Ostaszewski square HOD Strongly Luzin set middle diamond countably metacompact Shelah's Strong Hypothesis Minimal Walks Subtle cardinal Strong coloring Martin's Axiom Non-saturation Greatly Mahlo Generalized descriptive set theory Forcing Axioms Analytic sets Cohen real positive partition relation Hedetniemi's conjecture SNR regressive Souslin tree approachability ideal incompactness sap Subadditive free Boolean algebra Filter reflection Chang's conjecture Singular Density Almost countably chromatic Sakurai's Bell inequality Large Cardinals Fodor-type reflection OCA Commutative cancellative semigroups Sierpinski's onto mapping principle Nonspecial tree Reflecting stationary set square principles weak square Chromatic number Hereditarily Lindelöf space Antichain free Souslin tree Local Club Condensation. Vanishing levels Subnormal ideal Square-Brackets Partition Relations Amenable C-sequence nonmeager set club_AD Ascent Path specializable Souslin tree Selective Ultrafilter Well-behaved magma Weakly compact cardinal O-space Reduced Power Aronszajn tree Closed coloring Prikry-type forcing square Sigma-Prikry Rainbow sets Fast club Jonsson cardinal 54G20 GMA Parameterized proxy principle Microscopic Approach stationary hitting Ineffable cardinal Mandelbrot set higher Baire space reflection principles Almost-disjoint family Luzin set weak diamond PFA(S)[S] Foundations Singular cofinality Diamond Precaliber Subtle tree property Dushnik-Miller Successor of Singular Cardinal L-space Knaster Absoluteness Fat stationary set S-Space Diamond for trees Rock n' Roll weak Kurepa tree P-Ideal Dichotomy Partition Relations transformations b-scale Ulam matrix ccc super-Souslin tree Poset
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
Posted in Preprints, Souslin Hypothesis
Tagged C-sequence, free Souslin tree, Parameterized proxy principle, Souslin Tree, specializable Souslin tree, square principles, xbox
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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, 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