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