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Singular Density Diamond for trees diamond star Cardinal function Reduced Power Strong coloring Subadditive Erdos-Hajnal graphs Almost-disjoint family Aronszajn tree projective Boolean algebra AIM forcing Ulam matrix Selective Ultrafilter Fast club stationary hitting Forcing Vanishing levels transformations Microscopic Approach PFA(S)[S] Knaster weak square Foundations Distributive tree regressive Souslin tree Greatly Mahlo Successor of Regular Cardinal Chang's conjecture Ramsey theory over partitions Shelah's Strong Hypothesis sap Iterated forcing Constructible Universe Closed coloring Open Access Reflecting stationary set Sierpinski's onto mapping principle Ineffable cardinal middle diamond P-Ideal Dichotomy S-Space Prikry-type forcing Local Club Condensation. Forcing Axioms GMA tensor product graph club_AD polarized partition relation Uniformly homogeneous Subtle tree property incompactness Fat stationary set 54G20 unbounded function Souslin Tree weak diamond Parameterized proxy principle Subnormal ideal weak Kurepa tree Poset Mandelbrot set Martin's Axiom Coherent tree xbox Generalized descriptive set theory Minimal Walks Sakurai's Bell inequality super-Souslin tree Prevalent singular cardinals square Small forcing Almost countably chromatic PFA higher Baire space ZFC construction Dowker space Kurepa Hypothesis very good scale HOD Singular cofinality Almost Souslin Hereditarily Lindelöf space Generalized Clubs Cohen real positive partition relation O-space Rado's conjecture Amenable C-sequence Was Ulam right Absoluteness Lipschitz reduction Non-saturation SNR Analytic sets Partition Relations free Boolean algebra Nonspecial tree Commutative cancellative semigroups Sigma-Prikry Chromatic number Universal Sequences Club Guessing Antichain Whitehead Problem Erdos Cardinal coloring number full tree Luzin set Square-Brackets Partition Relations Singular cardinals combinatorics Diamond-sharp b-scale Cardinal Invariants countably metacompact Knaster and friends square principles Rainbow sets Postprocessing function C-sequence indecomposable ultrafilter Weakly compact cardinal Axiom R OCA nonmeager set strongly bounded groups Uniformly coherent Uniformization Diamond Jonsson cardinal stick Hindman's Theorem Well-behaved magma Slim tree Hedetniemi's conjecture Successor of Singular Cardinal Strongly Luzin set L-space Large Cardinals Ascent Path Precaliber Fodor-type reflection stationary reflection Rock n' Roll ccc approachability ideal Filter reflection reflection principles free Souslin tree Subtle cardinal Dushnik-Miller Ostaszewski square specializable Souslin tree
Author Archives: Assaf Rinot
Sigma-Prikry forcing III: Down to Aleph_omega
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_\omega$ together with the reflection of all stationary subsets of $\aleph_{\omega+1}$. This shows that two classical results of … Continue reading
Was Ulam right? I: Basic theory and subnormal ideals
Joint work with Tanmay Inamdar. Abstract. We introduce various coloring principles which generalize the so-called onto mapping principle of Sierpinski to larger cardinals and general ideals. We prove that these principles capture the notion of an Ulam matrix and allow … Continue reading
Knaster and friends III: Subadditive colorings
Joint work with Chris Lambie-Hanson. Abstract. We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa$, the existence … Continue reading
Posted in Partition Relations, Publications
Tagged Ascent Path, Knaster and friends, Open Access, P-Ideal Dichotomy, sap, square, Subadditive, Uniformly coherent
1 Comment
Strongest transformations
Joint work with Jing Zhang. Abstract. We continue our study of maps transforming high-dimensional complicated objects into squares of stationary sets. Previously, we proved that many such transformations exist in ZFC, and here we address the consistency of the strongest … Continue reading
Posted in Partition Relations, Publications
Tagged Diamond, Minimal Walks, square, Square-Brackets Partition Relations, stick, transformations, xbox
2 Comments
On the ideal J[kappa]
Abstract. Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[\kappa]$, from which we confirm that $\kappa$-Souslin trees exist in various models of interest. As a corollary we … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged Cardinal Invariants, Cohen real, nonmeager set
1 Comment
Ramsey theory over partitions III: Strongly Luzin sets and partition relations
Joint work with Menachem Kojman and Juris Steprāns. Abstract. The strongest type of coloring of pairs of countable ordinals, gotten by Todorcevic from a strongly Luzin set, is shown to be equivalent to the existence of a nonmeager set of … Continue reading
Ramsey theory over partitions I: Positive Ramsey relations from forcing axioms
Joint work with Menachem Kojman and Juris Steprāns. Abstract. In this series of papers, we advance Ramsey theory of colorings over partitions. In this part, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions … Continue reading
Posted in Partition Relations, Publications
Tagged 03E02, 03E17, 03E35, GMA, Martin's Axiom, positive partition relation, Ramsey theory over partitions
1 Comment
A guessing principle from a Souslin tree, with applications to topology
Joint work with Roy Shalev. Abstract. We introduce a new combinatorial principle which we call $\clubsuit_{AD}$. This principle asserts the existence of a certain multi-ladder system with guessing and almost-disjointness features, and is shown to be sufficient for carrying out … Continue reading
Posted in Publications, Souslin Hypothesis, Topology
Tagged club_AD, Dowker space, O-space, regressive Souslin tree, S-Space, Souslin Tree, Vanishing levels
2 Comments
MFO workshop in Set Theory, April 2020
The Set Theory workshop in Obwerwolfach was supposed to take place April 2020, but was transformed into a webinar, due to COVID-19. Here you will find the title, abstract, slides and video of my webinar talk. Talk Title: Transformations of … Continue reading
Posted in Invited Talks
Comments Off on MFO workshop in Set Theory, April 2020
Transformations of the transfinite plane
Joint work with Jing Zhang. Abstract. We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every … Continue reading
