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Strongly Luzin set Jonsson cardinal diamond star approachability ideal Analytic sets weak Kurepa tree Singular cofinality Uniformly homogeneous square principles Ineffable cardinal Whitehead Problem Forcing Fat stationary set regressive Souslin tree ccc Subtle cardinal Was Ulam right? stationary reflection Subnormal ideal Constructible Universe Distributive tree weak square Hedetniemi's conjecture Reflecting stationary set Commutative projection system Mandelbrot set Uniformly coherent Greatly Mahlo 54G20 Partition Relations unbounded function Small forcing Intersection model Dushnik-Miller polarized partition relation Sigma-Prikry Reduced Power very good scale Iterated forcing middle diamond Selective Ultrafilter Fodor-type reflection free Boolean algebra Souslin Tree indecomposable filter HOD Cardinal Invariants Coherent tree Local Club Condensation. Hereditarily Lindelöf space Rainbow sets Closed coloring countably metacompact PFA Sierpinski's onto mapping principle Large Cardinals C-sequence AIM forcing Erdos-Hajnal graphs L-space Cardinal function Club Guessing Generalized Clubs Commutative cancellative semigroups Respecting tree Antichain OCA nonmeager set Prikry-type forcing strongly bounded groups Forcing with side conditions Generalized descriptive set theory PFA(S)[S] sap Chromatic number Singular cardinals combinatorics Amenable C-sequence Lipschitz reduction Square-Brackets Partition Relations Countryman line Partition relations for trees super-Souslin tree Strong coloring SNR Diamond P-Ideal Dichotomy Knaster weak diamond Filter reflection projective Boolean algebra specializable Souslin tree Precaliber club_AD GMA Well-behaved magma Chang's conjecture incompactness Minimal Walks Forcing Axioms Almost-disjoint family Erdos Cardinal Vanishing levels S-Space Sakurai's Bell inequality Ascending path higher Baire space Dowker space Luzin set square Microscopic Approach Monotonically far Diamond-sharp Rock n' Roll Successor of Singular Cardinal stationary hitting stick Prevalent singular cardinals reflection principles coloring number Hindman's Theorem full tree Aronszajn tree Ostaszewski square Subtle tree property Absoluteness Nonspecial tree Almost Souslin Uniformization Interval topology on trees tensor product graph Poset Slim tree xbox perfectly normal b-scale Subadditive transformations Shelah's Strong Hypothesis Axiom R Strongly compact cardinal Fast club Universal Sequences Entangled linear order Ramsey theory over partitions Open Access Knaster and friends O-space Rado's conjecture Non-saturation Diamond for trees Foundations free Souslin tree Almost countably chromatic Weakly compact cardinal Singular Density positive partition relation Successor of Regular Cardinal Cohen real Martin's Axiom Ulam matrix Parameterized proxy principle ZFC construction Ascent Path Postprocessing function Kurepa Hypothesis
Author Archives: Assaf Rinot
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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Set Theory and its Applications in Topology, September 2016
I gave an invited talk at the Set Theory and its Applications in Topology meeting, Oaxaca, September 11-16, 2016. The talk was on the $\aleph_2$-Souslin problem. If you are interested in seeing the effect of a jet lag, the video is … Continue reading
Strong failures of higher analogs of Hindman’s Theorem
Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading
Posted in Groups, Partition Relations, Publications
Tagged 03E02, 03E35, 03E75, 05A17, 05D10, 11P99, 20M14, Chang's conjecture, Commutative cancellative semigroups, Entangled linear order, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal, ZFC construction
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More notions of forcing add a Souslin tree
Joint work with Ari Meir Brodsky. Abstract. An $\aleph_1$-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But 15 years after Tennenbaum and independently Jech devised notions of forcing for introducing … Continue reading
