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