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