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