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Diamond b-scale Generalized descriptive set theory Dowker space higher Baire space diamond star Square-Brackets Partition Relations Weakly compact cardinal Well-behaved magma Singular Density Ascending path Intersection model Chang's conjecture Distributive tree Vanishing levels Erdos Cardinal Hereditarily Lindelöf space polarized partition relation Iterated forcing OCA Partition relations for trees Constructible Universe Almost countably chromatic O-space Cardinal function Singular cardinals combinatorics Knaster Rock n' Roll xbox Cardinal Invariants Prevalent singular cardinals SNR Reduced Power Nonspecial tree approachability ideal Souslin Tree Forcing with side conditions C-sequence indecomposable filter Was Ulam right? unbounded function weak Kurepa tree Ascent Path projective Boolean algebra middle diamond Partition Relations GMA Fat stationary set Dushnik-Miller Uniformization PFA weak square square principles Successor of Regular Cardinal Martin's Axiom Closed coloring Hedetniemi's conjecture PFA(S)[S] Commutative cancellative semigroups free Boolean algebra stationary reflection Respecting tree Monotonically far Knaster and friends Fast club Diamond for trees Absoluteness Cohen real strongly bounded groups Ulam matrix super-Souslin tree Kurepa Hypothesis Postprocessing function Open Access L-space Successor of Singular Cardinal nonmeager set P-Ideal Dichotomy Microscopic Approach Slim tree Generalized Clubs Coherent tree Singular cofinality Rado's conjecture 54G20 tensor product graph Countryman line Sierpinski's onto mapping principle Large Cardinals sap Forcing very good scale stick Interval topology on trees perfectly normal incompactness countably metacompact Filter reflection Hindman's Theorem Non-saturation positive partition relation Precaliber free Souslin tree regressive Souslin tree Ostaszewski square reflection principles full tree Sigma-Prikry Mandelbrot set Ineffable cardinal transformations Almost Souslin Strongly Luzin set Strongly compact cardinal Club Guessing Minimal Walks Universal Sequences Local Club Condensation. square stationary hitting Aronszajn tree Reflecting stationary set HOD Poset Whitehead Problem Sakurai's Bell inequality Greatly Mahlo Axiom R Small forcing ccc Fodor-type reflection AIM forcing Entangled linear order weak diamond Amenable C-sequence Rainbow sets S-Space Parameterized proxy principle Antichain Analytic sets club_AD ZFC construction Strong coloring Shelah's Strong Hypothesis Subtle tree property Forcing Axioms Selective Ultrafilter Subadditive Chromatic number Diamond-sharp Subtle cardinal Almost-disjoint family Uniformly coherent Erdos-Hajnal graphs Subnormal ideal Prikry-type forcing Ramsey theory over partitions Luzin set coloring number Foundations Commutative projection system specializable Souslin tree Jonsson cardinal Uniformly homogeneous Lipschitz reduction
Tag Archives: Diamond
Diamond on Kurepa trees
Joint work with Ziemek Kostana and Saharon Shelah. Abstract. We introduce a new weak variation of diamond that is meant to guess only the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by … 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
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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
Weak square and stationary reflection
Joint work with Gunter Fuchs. Abstract. It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, stationary reflection, weak square
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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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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
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Putting a diamond inside the square
Abstract. By a 35-year-old theorem of Shelah, $\square_\lambda+\diamondsuit(\lambda^+)$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$. Here, it is proved that $\square_\lambda+\diamondsuit(\lambda^+)$ is equivalent to square-with-built-in-diamond_lambda for every singular cardinal $\lambda$. Downloads: Citation information: A. Rinot, Putting a diamond inside … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Diamond, square, Successor of Singular Cardinal
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Many diamonds from just one
Recall Jensen’s diamond principle over a stationary subset $S$ of a regular uncountable cardinal $\kappa$: there exists a sequence $\langle A_\alpha\mid \alpha\in S \rangle$ such that $\{\alpha\in S\mid A\cap\alpha=A_\alpha\}$ is stationary for every $A\subseteq\kappa$. Equivalently, there exists a sequence $\langle … Continue reading
Variations on diamond
Jensen’s diamond principle has many equivalent forms. The translation between these forms is often straight-forward, but there is one form whose equivalence to the usual form is somewhat surprising, and Devlin’s translation from one to the other, seems a little … Continue reading
