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Ostaszewski square ZFC construction super-Souslin tree Kurepa Hypothesis Rainbow sets Sakurai's Bell inequality Coherent tree Singular cardinals combinatorics strongly bounded groups Non-saturation Universal Sequences Diamond for trees Successor of Singular Cardinal SNR Cardinal function Souslin Tree indecomposable ultrafilter Foundations L-space Subnormal ideal Dushnik-Miller Hedetniemi's conjecture countably metacompact Diamond Uniformly homogeneous Club Guessing Cardinal Invariants Ineffable cardinal Commutative cancellative semigroups Prikry-type forcing b-scale polarized partition relation Aronszajn tree Successor of Regular Cardinal Weakly compact cardinal Hereditarily Lindelöf space Well-behaved magma Subtle tree property Partition Relations Knaster Square-Brackets Partition Relations Small forcing Chromatic number positive partition relation Shelah's Strong Hypothesis Fat stationary set Greatly Mahlo HOD Rado's conjecture C-sequence Filter reflection Minimal Walks free Boolean algebra approachability ideal Cohen real Almost Souslin square S-Space Luzin set AIM forcing Subadditive Postprocessing function weak Kurepa tree Slim tree Ascent Path Reduced Power Microscopic Approach weak diamond Amenable C-sequence Open Access Parameterized proxy principle Erdos Cardinal Fodor-type reflection PFA Axiom R Was Ulam right P-Ideal Dichotomy full tree Chang's conjecture Almost countably chromatic Nonspecial tree square principles Subtle cardinal O-space higher Baire space stick very good scale Selective Ultrafilter Prevalent singular cardinals Analytic sets Almost-disjoint family nonmeager set Sierpinski's onto mapping principle Strongly Luzin set tensor product graph Strong coloring 54G20 Local Club Condensation. Hindman's Theorem OCA transformations Martin's Axiom Precaliber Vanishing levels Singular cofinality Large Cardinals Closed coloring Lipschitz reduction Forcing Distributive tree Singular Density Generalized descriptive set theory Ramsey theory over partitions Jonsson cardinal stationary reflection reflection principles Erdos-Hajnal graphs GMA Rock n' Roll projective Boolean algebra diamond star Poset Forcing Axioms Ulam matrix sap Sigma-Prikry Reflecting stationary set coloring number ccc Diamond-sharp PFA(S)[S] unbounded function Generalized Clubs Mandelbrot set Uniformly coherent Antichain specializable Souslin tree xbox Whitehead Problem stationary hitting Knaster and friends Iterated forcing free Souslin tree regressive Souslin tree Constructible Universe Absoluteness Fast club incompactness Dowker space Uniformization middle diamond club_AD weak square
Tag Archives: square
Higher Souslin trees and the GCH, revisited
Abstract. It is proved that for every uncountable cardinal $\lambda$, GCH+$\square(\lambda^+)$ entails the existence of a $\text{cf}(\lambda)$-complete $\lambda^+$-Souslin tree. In particular, if GCH holds and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in Godel’s constructible universe, improving … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, Open Access, regressive Souslin tree, Souslin Tree, square, Weakly compact cardinal, xbox
16 Comments
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
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, 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
Chain conditions of products, and weakly compact cardinals
Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading
Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
2 Comments
Square principles
Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle … Continue reading
The order-type of clubs in a square sequence
Recall Jensen’s notion of square: Definition (Jensen): For an infinite cardinal $\lambda$, $\square_\lambda$ asserts the existence of a sequence $\overrightarrow C=\left\langle C_\alpha\mid\alpha\in\text{acc}(\lambda^+)\right\rangle$ such that for every limit $\alpha<\lambda^+$: $C_\alpha$ is a club subset of $\alpha$ of order-type $\le\lambda$; if $\beta\in\text{acc}(C_\alpha)$, … Continue reading
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
Young Researchers in Set Theory, March 2011
These are the slides of a talk I gave at the Young Researchers in Set Theory 2011 meeting (Königswinter, 21–25 March 2011). Talk Title: Around Jensen’s square principle Abstract: Jensen‘s square principle for a cardinal $\lambda$ asserts the existence of a particular ladder … Continue reading
The failure of diamond on a reflecting stationary set
Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with … Continue reading
