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