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countably metacompact Partition Relations free Souslin tree Uniformly coherent Cardinal function Universal Sequences Martin's Axiom Axiom R Prikry-type forcing Sakurai's Bell inequality Large Cardinals Ascent Path Generalized descriptive set theory Interval topology on trees Diamond L-space Precaliber positive partition relation Rado's conjecture weak diamond Kurepa Hypothesis ZFC construction higher Baire space GMA Sigma-Prikry Singular Density square principles approachability ideal P-Ideal Dichotomy incompactness Commutative cancellative semigroups Forcing strongly bounded groups Strongly compact cardinal Sierpinski's onto mapping principle Non-saturation Ineffable cardinal Greatly Mahlo Distributive tree transformations Constructible Universe Cohen real Prevalent singular cardinals sap Almost Souslin Selective Ultrafilter Partition relations for trees reflection principles Diamond for trees Absoluteness Analytic sets Chromatic number Filter reflection PFA Slim tree diamond star Ostaszewski square weak square Almost countably chromatic Forcing Axioms b-scale Small forcing Reflecting stationary set square Dushnik-Miller stationary reflection Foundations Nonspecial tree Hedetniemi's conjecture Fast club Minimal Walks Amenable C-sequence club_AD Open Access Erdos Cardinal Strong coloring perfectly normal Uniformization Antichain Lipschitz reduction HOD coloring number weak Kurepa tree regressive Souslin tree Knaster and friends unbounded function Hindman's Theorem super-Souslin tree Almost-disjoint family Knaster very good scale S-Space nonmeager set Microscopic Approach Successor of Singular Cardinal Reduced Power Aronszajn tree Ulam matrix Uniformly homogeneous Subtle tree property Forcing with side conditions indecomposable filter middle diamond Respecting tree projective Boolean algebra Chang's conjecture Strongly Luzin set Monotonically far Ascending path Well-behaved magma Jonsson cardinal PFA(S)[S] Weakly compact cardinal ccc Intersection model Subnormal ideal Was Ulam right? Vanishing levels Generalized Clubs Erdos-Hajnal graphs Mandelbrot set Closed coloring Club Guessing C-sequence Commutative projection system Local Club Condensation. OCA free Boolean algebra Shelah's Strong Hypothesis Countryman line full tree Poset Rock n' Roll Square-Brackets Partition Relations SNR Cardinal Invariants Singular cofinality Subadditive Ramsey theory over partitions Parameterized proxy principle Singular cardinals combinatorics Fodor-type reflection Fat stationary set tensor product graph 54G20 Iterated forcing Coherent tree O-space Entangled linear order stick polarized partition relation Hereditarily Lindelöf space Souslin Tree specializable Souslin tree Postprocessing function Diamond-sharp xbox Rainbow sets Dowker space AIM forcing Successor of Regular Cardinal Luzin set Subtle cardinal stationary hitting Whitehead Problem
Author Archives: Assaf Rinot
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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The 14th International Workshop on Set Theory in Luminy, October 2017
I gave an invited talk at the 14th International Workshop on Set Theory in Luminy in Marseille, October 2017. Talk Title: Distributive Aronszajn trees Abstract: It is well-known that that the statement “all $\aleph_1$-Aronszajn trees are special” is consistent with ZFC … Continue reading
