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