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