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Kurepa Hypothesis Ramsey theory over partitions sap O-space Subtle cardinal Prikry-type forcing reflection principles Diamond for trees Intersection model weak diamond Chang's conjecture Ascent Path stationary hitting stick Diamond-sharp Axiom R Jonsson cardinal SNR Uniformly coherent free Souslin tree diamond star Almost-disjoint family club_AD Subnormal ideal P-Ideal Dichotomy Weakly compact cardinal Small forcing Local Club Condensation. Slim tree Square-Brackets Partition Relations Cohen real Absoluteness Fat stationary set Reduced Power Foundations Mandelbrot set Almost countably chromatic Whitehead Problem countably metacompact Closed coloring Interval topology on trees Club Guessing projective Boolean algebra approachability ideal Fast club tensor product graph Vanishing levels Successor of Regular Cardinal free Boolean algebra Strongly compact cardinal Respecting tree Knaster and friends AIM forcing Rado's conjecture unbounded function square GMA Dushnik-Miller Strongly Luzin set Singular cofinality Constructible Universe Minimal Walks Erdos Cardinal incompactness Lipschitz reduction Monotonically far Hindman's Theorem Commutative cancellative semigroups PFA strongly bounded groups coloring number PFA(S)[S] higher Baire space Strong coloring Subadditive ccc very good scale Sierpinski's onto mapping principle Universal Sequences Generalized descriptive set theory Martin's Axiom Amenable C-sequence super-Souslin tree S-Space Uniformly homogeneous transformations Hereditarily Lindelöf space Filter reflection Souslin Tree square principles Precaliber weak Kurepa tree Commutative projection system OCA Analytic sets Ascending path weak square Microscopic Approach Knaster perfectly normal Forcing Axioms Uniformization Generalized Clubs polarized partition relation Large Cardinals Subtle tree property Almost Souslin nonmeager set Luzin set Ulam matrix Postprocessing function Aronszajn tree Cardinal function Ostaszewski square C-sequence HOD Erdos-Hajnal graphs Successor of Singular Cardinal Greatly Mahlo Prevalent singular cardinals Countryman line Was Ulam right? Hedetniemi's conjecture L-space Coherent tree Shelah's Strong Hypothesis Well-behaved magma 54G20 Forcing with side conditions Forcing Nonspecial tree Open Access Entangled linear order Chromatic number full tree indecomposable filter Iterated forcing Antichain Reflecting stationary set Sakurai's Bell inequality positive partition relation Diamond Sigma-Prikry Rainbow sets Ineffable cardinal Parameterized proxy principle middle diamond Singular cardinals combinatorics regressive Souslin tree Rock n' Roll Fodor-type reflection b-scale Dowker space Singular Density Cardinal Invariants Poset xbox stationary reflection Non-saturation Selective Ultrafilter Partition Relations Partition relations for trees ZFC construction Distributive tree specializable Souslin tree
Category Archives: Squares and Diamonds
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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Jensen’s diamond principle and its relatives
This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … Continue reading
