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O-space Kurepa Hypothesis Jonsson cardinal Commutative projection system Chang's conjecture Erdos-Hajnal graphs Diamond for trees Subadditive Vanishing levels Knaster and friends PFA(S)[S] coloring number xbox Well-behaved magma Large Cardinals Reflecting stationary set unbounded function Entangled linear order full tree Strong coloring Souslin Tree Coherent tree middle diamond Partition relations for trees Knaster Non-saturation Sigma-Prikry Nonspecial tree Forcing Axioms square principles Diamond-sharp Reduced Power Greatly Mahlo Small forcing square incompactness polarized partition relation PFA Absoluteness Lipschitz reduction positive partition relation specializable Souslin tree Singular cardinals combinatorics Postprocessing function Interval topology on trees S-Space Uniformly homogeneous weak square Successor of Singular Cardinal Fat stationary set Shelah's Strong Hypothesis Aronszajn tree Erdos Cardinal regressive Souslin tree Filter reflection Chromatic number L-space Generalized descriptive set theory Dushnik-Miller countably metacompact Minimal Walks Intersection model Uniformly coherent Ascent Path Successor of Regular Cardinal ZFC construction club_AD Forcing with side conditions Open Access Almost-disjoint family stationary reflection Antichain P-Ideal Dichotomy Whitehead Problem Slim tree C-sequence Fodor-type reflection Ulam matrix perfectly normal Sakurai's Bell inequality Strongly Luzin set Hereditarily Lindelöf space Distributive tree Precaliber Club Guessing Strongly compact cardinal Ineffable cardinal nonmeager set Was Ulam right? free Souslin tree Subnormal ideal indecomposable filter approachability ideal Axiom R Prikry-type forcing Cardinal function Monotonically far Square-Brackets Partition Relations Ascending path Parameterized proxy principle Rainbow sets Ostaszewski square Singular cofinality Ramsey theory over partitions Singular Density AIM forcing Almost Souslin stationary hitting strongly bounded groups Dowker space GMA Prevalent singular cardinals weak Kurepa tree OCA transformations higher Baire space Generalized Clubs tensor product graph projective Boolean algebra Forcing b-scale Almost countably chromatic Fast club Local Club Condensation. very good scale Hindman's Theorem Uniformization stick free Boolean algebra diamond star 54G20 Partition Relations Microscopic Approach Commutative cancellative semigroups Diamond Rock n' Roll SNR Rado's conjecture super-Souslin tree Hedetniemi's conjecture Countryman line sap Selective Ultrafilter ccc Subtle cardinal Foundations Mandelbrot set Respecting tree Iterated forcing Cardinal Invariants Poset weak diamond Martin's Axiom Sierpinski's onto mapping principle Constructible Universe Analytic sets Cohen real Universal Sequences Subtle tree property Closed coloring Amenable C-sequence HOD Luzin set reflection principles Weakly compact cardinal
Author Archives: Assaf Rinot
Forcing and its Applications Retrospective Workshop, April 2015
I gave an invited talk at Forcing and its Applications Retrospective Workshop, Toronto, April 1st, 2015. Title: A microscopic approach to Souslin trees constructions Abstract: We present an approach to construct $\kappa$-Souslin trees that is insensitive to the identity of … Continue reading
Posted in Invited Talks
Tagged Microscopic Approach, Parameterized proxy principle, Souslin Tree
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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
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Generalizations of Martin’s Axiom and the well-met condition
Recall that Martin’s Axiom asserts that for every partial order $\mathbb P$ satisfying c.c.c., and for any family $\mathcal D$ of $<2^{\aleph_0}$ many dense subsets of $\mathbb P$, there exists a directed subset $G$ of $\mathbb P$ such that $G\cap … Continue reading
Posted in Blog, Expository
Tagged ccc, Forcing Axioms, GMA, Martin's Axiom, Uniformization
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Many diamonds from just one
Recall Jensen’s diamond principle over a stationary subset $S$ of a regular uncountable cardinal $\kappa$: there exists a sequence $\langle A_\alpha\mid \alpha\in S \rangle$ such that $\{\alpha\in S\mid A\cap\alpha=A_\alpha\}$ is stationary for every $A\subseteq\kappa$. Equivalently, there exists a sequence $\langle … Continue reading
Same Graph, Different Universe
Abstract. May the same graph admit two different chromatic numbers in two different universes? how about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Godel’s constructible … Continue reading
Posted in Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, approachability ideal, Chromatic number, Constructible Universe, Forcing, Ostaszewski square
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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
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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
INFTY Final Conference, March 2014
I gave an invited talk at the INFTY Final Conference meeting, Bonn, March 4-7, 2014. [Curiosity: Georg Cantor was born March 3, 1845] Title: Same Graph, Different Universe. Abstract: In a paper from 1998, answering a question of Hajnal, Soukup … Continue reading
Partitioning the club guessing
In a recent paper, I am making use of the following fact. Theorem (Shelah, 1997). Suppose that $\kappa$ is an accessible cardinal (i.e., there exists a cardinal $\theta<\kappa$ such that $2^\theta\ge\kappa)$. Then there exists a sequence $\langle g_\delta:C_\delta\rightarrow\omega\mid \delta\in E^{\kappa^+}_\kappa\rangle$ … Continue reading
