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square principles Successor of Regular Cardinal Diamond-sharp Subtle tree property Subtle cardinal Dushnik-Miller 54G20 Cardinal function PFA(S)[S] indecomposable ultrafilter Vanishing levels Fodor-type reflection Ineffable cardinal OCA Strongly Luzin set Microscopic Approach Ascent Path Iterated forcing Singular cofinality weak diamond Club Guessing middle diamond Ramsey theory over partitions Was Ulam right? Generalized Clubs Uniformly coherent Nonspecial tree Axiom R HOD L-space Ulam matrix transformations Erdos Cardinal Coherent tree Open Access full tree Strongly compact cardinal Intersection model Fast club Diamond for trees Cohen real PFA Weakly compact cardinal unbounded function Prikry-type forcing Universal Sequences Well-behaved magma Fat stationary set Greatly Mahlo SNR Mandelbrot set Local Club Condensation. Subnormal ideal Reduced Power weak square Countryman line Antichain Absoluteness Whitehead Problem Aronszajn tree Knaster and friends strongly bounded groups positive partition relation regressive Souslin tree Sierpinski's onto mapping principle Hedetniemi's conjecture Forcing Axioms Subadditive Rado's conjecture diamond star Postprocessing function super-Souslin tree Filter reflection coloring number very good scale Ostaszewski square Precaliber Singular cardinals combinatorics Commutative projection system tensor product graph countably metacompact AIM forcing Singular Density Kurepa Hypothesis Rainbow sets Amenable C-sequence Hereditarily Lindelöf space Closed coloring Minimal Walks Small forcing Sakurai's Bell inequality Diamond O-space Commutative cancellative semigroups stationary hitting Shelah's Strong Hypothesis Slim tree Respecting tree Cardinal Invariants C-sequence Parameterized proxy principle Analytic sets weak Kurepa tree specializable Souslin tree Generalized descriptive set theory Almost Souslin Chromatic number sap xbox Uniformization reflection principles stick free Souslin tree Almost countably chromatic Partition Relations Luzin set square Strong coloring ZFC construction free Boolean algebra P-Ideal Dichotomy Constructible Universe Square-Brackets Partition Relations higher Baire space Sigma-Prikry Successor of Singular Cardinal Large Cardinals Jonsson cardinal Lipschitz reduction stationary reflection Non-saturation Rock n' Roll Hindman's Theorem incompactness polarized partition relation projective Boolean algebra Selective Ultrafilter Knaster ccc Uniformly homogeneous Forcing Souslin Tree Martin's Axiom S-Space Chang's conjecture Foundations Prevalent singular cardinals Poset Distributive tree GMA approachability ideal Reflecting stationary set Almost-disjoint family b-scale Erdos-Hajnal graphs club_AD nonmeager set Dowker space
Author Archives: Assaf Rinot
Inclusion modulo nonstationary
Joint work with Gabriel Fernandes and Miguel Moreno. Abstract. A classical theorem of Hechler asserts that the structure $\left(\omega^\omega,\le^*\right)$ is universal in the sense that for any $\sigma$-directed poset $\mathbb P$ with no maximal element, there is a ccc forcing … Continue reading
50 Years of Set Theory in Toronto, May 2019
I gave an invited talk at the 50 Years of Set Theory in Toronto meeting, Fields Institute for Research in Mathematical Sciences, May 2019. Talk Title: Analytic quasi-orders and two forms of diamond Abstract: We study Borel reduction of equivalence relations … Continue reading
Posted in Invited Talks
Tagged Analytic sets, Diamond-sharp, Generalized descriptive set theory
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Partitioning a reflecting stationary set
Joint work with Maxwell Levine. Abstract. We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinals combinatorics, we infer … Continue reading
4th Arctic Set Theory Workshop, January 2019
I gave an invited talk at the Arctic Set Theory Workshop 4 in Kilpisjärvi, January 2019. Talk Title: Splitting a stationary set: Is there another way? Abstract: Motivated by a problem in pcf theory, we seek for a new way … Continue reading
Posted in Invited Talks
Tagged Club Guessing, Reflecting stationary set, Ulam matrix, very good scale
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Souslin trees at successors of regular cardinals
Abstract. We present a weak sufficient condition for the existence of Souslin trees at successor of regular cardinals. The result is optimal and simultaneously improves an old theorem of Gregory and a more recent theorem of the author. Downloads: Citation … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged Parameterized proxy principle, Souslin Tree
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11th Young Set Theory Workshop, June 2018
I gave a 4-lecture tutorial at the 11th Young Set Theory Workshop, Lausanne, June 2018. Title: In praise of C-sequences. Abstract. Ulam and Solovay showed that any stationary set may be split into two. Is it also the case that … Continue reading
Posted in Invited Talks
Tagged Aronszajn tree, C-sequence, incompactness, Knaster, Minimal Walks, Postprocessing function, square
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Knaster and friends I: Closed colorings and precalibers
Joint work with Chris Lambie-Hanson. Abstract. The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $\kappa$-cc posets whose squares … Continue reading
A remark on Schimmerling’s question
Joint work with Ari Meir Brodsky. Abstract. Schimmerling asked whether $\square^*_\lambda$ together with GCH entails the existence of a $\lambda^+$-Souslin tree, for a singular cardinal $\lambda$. Here, we provide an affirmative answer under the additional assumption that there exists a … Continue reading
Weak square and stationary reflection
Joint work with Gunter Fuchs. Abstract. It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, stationary reflection, weak square
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A strong form of König’s lemma
A student proposed to me the following strong form of König’s lemma: Conjecture. Suppose that $G=(V,E)$ is a countable a graph, and there is a partition of $V$ into countably many pieces $V=\bigcup_{n<\omega}V_n$, such that: for all $n<\omega$, $V_n$ is … Continue reading
Posted in Blog
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