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