Archives
Keywords
Reflecting stationary set Vanishing levels Uniformization stationary reflection Shelah's Strong Hypothesis OCA weak square projective Boolean algebra Jonsson cardinal Chromatic number Aronszajn tree Knaster Was Ulam right Sigma-Prikry Universal Sequences Dushnik-Miller Diamond-sharp Cardinal function Antichain coloring number diamond star Well-behaved magma Generalized Clubs Ulam matrix Forcing Axioms Foundations Singular cardinals combinatorics Strong coloring Closed coloring Hedetniemi's conjecture Absoluteness Whitehead Problem xbox Ascent Path Open Access Sierpinski's onto mapping principle stationary hitting stick very good scale Singular cofinality P-Ideal Dichotomy Reduced Power strongly bounded groups Prikry-type forcing 54G20 S-Space Fat stationary set tensor product graph incompactness Hereditarily Lindelöf space Cardinal Invariants regressive Souslin tree Chang's conjecture Constructible Universe super-Souslin tree Strongly Luzin set higher Baire space Erdos Cardinal Small forcing Hindman's Theorem Axiom R approachability ideal Selective Ultrafilter Almost countably chromatic Commutative cancellative semigroups Singular Density transformations weak diamond Mandelbrot set Iterated forcing Sakurai's Bell inequality Non-saturation Square-Brackets Partition Relations free Boolean algebra Parameterized proxy principle AIM forcing b-scale Analytic sets square Subtle tree property middle diamond Uniformly homogeneous club_AD C-sequence Rock n' Roll Amenable C-sequence Forcing Postprocessing function Generalized descriptive set theory Luzin set ZFC construction polarized partition relation SNR Erdos-Hajnal graphs O-space indecomposable ultrafilter Weakly compact cardinal PFA nonmeager set unbounded function Uniformly coherent full tree Successor of Regular Cardinal Ineffable cardinal Ostaszewski square countably metacompact Subtle cardinal square principles Diamond positive partition relation Souslin Tree Kurepa Hypothesis GMA Filter reflection Dowker space Fodor-type reflection Successor of Singular Cardinal Precaliber Minimal Walks Slim tree Prevalent singular cardinals HOD L-space Subadditive ccc weak Kurepa tree Fast club free Souslin tree Diamond for trees Greatly Mahlo Nonspecial tree Subnormal ideal Almost Souslin Coherent tree sap Club Guessing reflection principles Rado's conjecture Knaster and friends Cohen real Martin's Axiom specializable Souslin tree Lipschitz reduction Rainbow sets Microscopic Approach Distributive tree Ramsey theory over partitions Poset Partition Relations Almost-disjoint family PFA(S)[S] Local Club Condensation. Large Cardinals
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 only guess the branches of a Kurepa tree. We demonstrate that this variation is considerably weaker than diamond by … Continue reading
Posted in Preprints, Squares and Diamonds
Tagged Diamond, Diamond for trees, Iterated forcing, Kurepa Hypothesis, weak Kurepa tree
Comments Off on Diamond on Kurepa trees
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
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
Leave a comment
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
1 Comment
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
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
1 Comment
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
The search for diamonds
Abstract: This is a review I wrote for the Bulletin of Symbolic Logic on the following papers: Saharon Shelah, Middle Diamond, Archive for Mathematical Logic, vol. 44 (2005), pp. 527–560. Saharon Shelah, Diamonds, Proceedings of the American Mathematical Society, vol. … Continue reading
Posted in Publications, Reviews, Squares and Diamonds
Tagged Diamond, middle diamond, weak diamond, weak square
1 Comment