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Hindman's Theorem weak Kurepa tree unbounded function Reflecting stationary set Knaster Rainbow sets ccc Subtle cardinal club_AD approachability ideal Commutative cancellative semigroups higher Baire space Dowker space perfectly normal Chang's conjecture square diamond star Microscopic Approach Small forcing Cohen real Club Guessing regressive Souslin tree L-space Greatly Mahlo Was Ulam right? Sierpinski's onto mapping principle Rock n' Roll Well-behaved magma Large Cardinals Monotonically far SNR Diamond-sharp Selective Ultrafilter Almost countably chromatic xbox Fast club Forcing Axioms Parameterized proxy principle Local Club Condensation. Lipschitz reduction Amenable C-sequence GMA Closed coloring Vanishing levels Shelah's Strong Hypothesis Intersection model Diamond super-Souslin tree C-sequence full tree Universal Sequences Prikry-type forcing Hereditarily Lindelöf space Absoluteness Ineffable cardinal Knaster and friends coloring number Filter reflection polarized partition relation Uniformly coherent square principles Ramsey theory over partitions Aronszajn tree Precaliber Foundations Strongly compact cardinal Singular Density Axiom R Interval topology on trees Entangled linear order Rado's conjecture PFA(S)[S] Square-Brackets Partition Relations Respecting tree stationary hitting Erdos Cardinal strongly bounded groups 54G20 nonmeager set tensor product graph Non-saturation middle diamond Almost Souslin stick Singular cofinality Hedetniemi's conjecture Postprocessing function Whitehead Problem S-Space specializable Souslin tree Sakurai's Bell inequality Martin's Axiom O-space Analytic sets stationary reflection transformations countably metacompact HOD Strong coloring Kurepa Hypothesis incompactness P-Ideal Dichotomy Reduced Power Countryman line Fodor-type reflection Uniformly homogeneous very good scale Generalized descriptive set theory Uniformization Chromatic number free Souslin tree ZFC construction Cardinal function weak square Ulam matrix b-scale Subtle tree property projective Boolean algebra Souslin Tree Successor of Regular Cardinal Successor of Singular Cardinal Constructible Universe Partition Relations PFA Almost-disjoint family positive partition relation Diamond for trees Dushnik-Miller Weakly compact cardinal Ostaszewski square Sigma-Prikry OCA weak diamond Open Access Partition relations for trees Ascending path free Boolean algebra Iterated forcing Slim tree Strongly Luzin set Cardinal Invariants Generalized Clubs Coherent tree indecomposable filter Erdos-Hajnal graphs Subadditive Forcing with side conditions Antichain Luzin set Subnormal ideal Forcing Nonspecial tree Commutative projection system sap reflection principles Ascent Path Jonsson cardinal Distributive tree Minimal Walks Poset Prevalent singular cardinals Singular cardinals combinatorics Fat stationary set Mandelbrot set AIM forcing
Author Archives: Assaf Rinot
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
A forcing axiom deciding the generalized Souslin Hypothesis
Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
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6th European Set Theory Conference, July 2017
I gave a 3-lecture tutorial at the 6th European Set Theory Conference in Budapest, July 2017. Title: Strong colorings and their applications. Abstract. Consider the following questions. Is the product of two $\kappa$-cc partial orders again $\kappa$-cc? Does there exist … Continue reading
Posted in Invited Talks, Open Problems
Tagged b-scale, Cohen real, Luzin set, Minimal Walks, Souslin Tree, Square-Brackets Partition Relations
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Distributive Aronszajn trees
Joint work with Ari Meir Brodsky. Abstract. Ben-David and Shelah proved that if $\lambda$ is a singular strong-limit cardinal and $2^\lambda=\lambda^+$, then $\square^*_\lambda$ entails the existence of a $\lambda$-distributive $\lambda^+$-Aronszajn tree. Here, it is proved that the same conclusion remains … Continue reading
ASL North American Meeting, March 2017
I gave a plenary talk at the 2017 ASL North American Meeting in Boise, March 2017. Talk Title: The current state of the Souslin problem. Abstract: Recall that the real line is that unique separable, dense linear ordering with no endpoints in … Continue reading
MFO workshop in Set Theory, February 2017
I gave an invited talk at the Set Theory workshop in Obwerwolfach, February 2017. Talk Title: Coloring vs. Chromatic. Abstract: In a joint work with Chris Lambie-Hanson, we study the interaction between compactness for the chromatic number (of graphs) and … Continue reading
Posted in Invited Talks
Tagged Chromatic number, coloring number, incompactness, stationary reflection
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The eightfold way
Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading
Reflection on the coloring and chromatic numbers
Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Chang's conjecture, Chromatic number, coloring number, Fodor-type reflection, incompactness, Iterated forcing, Parameterized proxy principle, Postprocessing function, Rado's conjecture, square, stationary reflection
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