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