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Souslin Tree Almost Souslin HOD Successor of Singular Cardinal Diamond Luzin set club_AD Ramsey theory over partitions Ineffable cardinal Kurepa Hypothesis Reduced Power Singular cardinals combinatorics Diamond for trees SNR Fodor-type reflection Subtle tree property Uniformization Was Ulam right? Chang's conjecture Forcing Axioms diamond star Forcing with side conditions PFA Hindman's Theorem GMA Universal Sequences coloring number countably metacompact Nonspecial tree weak Kurepa tree Filter reflection unbounded function ccc Club Guessing square middle diamond Almost countably chromatic Erdos-Hajnal graphs PFA(S)[S] xbox Almost-disjoint family Amenable C-sequence tensor product graph Martin's Axiom free Boolean algebra Precaliber Cardinal Invariants Commutative cancellative semigroups b-scale Dushnik-Miller L-space Local Club Condensation. Forcing Analytic sets Rado's conjecture weak square Fast club Poset Closed coloring Distributive tree incompactness Knaster and friends Partition relations for trees Uniformly coherent Monotonically far Respecting tree Prevalent singular cardinals strongly bounded groups Partition Relations Vanishing levels higher Baire space projective Boolean algebra Dowker space Countryman line Minimal Walks ZFC construction Aronszajn tree Strongly Luzin set super-Souslin tree Subtle cardinal Selective Ultrafilter Small forcing Sierpinski's onto mapping principle Coherent tree Cohen real square principles C-sequence Commutative projection system Entangled linear order Rock n' Roll free Souslin tree polarized partition relation S-Space transformations Subadditive Chromatic number Singular cofinality Hereditarily Lindelöf space Successor of Regular Cardinal approachability ideal Prikry-type forcing Mandelbrot set Knaster Uniformly homogeneous Singular Density Foundations Subnormal ideal Constructible Universe stationary reflection reflection principles Ulam matrix Strong coloring Reflecting stationary set full tree regressive Souslin tree Hedetniemi's conjecture weak diamond Absoluteness Square-Brackets Partition Relations Lipschitz reduction OCA Shelah's Strong Hypothesis Slim tree perfectly normal Interval topology on trees Sakurai's Bell inequality very good scale Large Cardinals Postprocessing function Weakly compact cardinal Greatly Mahlo stick P-Ideal Dichotomy indecomposable filter stationary hitting Whitehead Problem Microscopic Approach Parameterized proxy principle Antichain Rainbow sets Ascending path 54G20 Erdos Cardinal Diamond-sharp Axiom R specializable Souslin tree Non-saturation Generalized Clubs positive partition relation Ascent Path Sigma-Prikry Open Access Cardinal function O-space Strongly compact cardinal Generalized descriptive set theory Ostaszewski square Iterated forcing Intersection model sap Jonsson cardinal Well-behaved magma AIM forcing nonmeager set Fat stationary set
Category Archives: Squares and Diamonds
More notions of forcing add a square
Joint work with Yair Hayut and Zhixing You. Abstract. Foreman and Magidor showed that the continuum hypothesis implies the existence of a countably-closed $\aleph_2$-cc forcing notion $\mathbb P$ for adding $\square_{\omega_1}$. Here, we show that $\mathbb P$ may consistently be … Continue reading
Posted in Squares and Diamonds
Tagged full tree, Souslin Tree, square principles, super-Souslin tree
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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
A club guessing toolbox I
Joint work with Tanmay Inamdar. Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove Shelah’s ZFC bound on the power of the first singular cardinal. These principles have … Continue reading
Inclusion modulo nonstationary
Joint work with Gabriel Fernandes and Miguel Moreno. Abstract. A classical theorem of Hechler asserts that the structure $\left(\omega^\omega,\le^*\right)$ is universal in the sense that for any $\sigma$-directed poset $\mathbb P$ with no maximal element, there is a ccc forcing … 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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Distributive Aronszajn trees
Joint work with Ari Meir Brodsky. Abstract. Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved that the same conclusion remains … Continue reading
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
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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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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
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