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Author Archives: Assaf Rinot
P.O.I. Workshop in pure and descriptive set theory, September 2015
I gave an invited talk at the P.O.I Workshop in pure and descriptive set theory, Torino, September 26, 2015. Title: $\aleph_3$-trees. Abstract: We inspect the constructions of four quite different $\aleph_3$-Souslin trees.
Reduced powers of Souslin trees
Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed … Continue reading
The Apter-Gitik birthday conference, May 2015
I give an invited (blackboard) talk at the Apter-Gitik birthday conference, Carnegie Mellon University, May 30-31 2015. Title: 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 … Continue reading
Posted in Invited Talks
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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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