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