Archives
Keywords
weak Kurepa tree Iterated forcing stick AIM forcing free Souslin tree Sakurai's Bell inequality Reflecting stationary set Uniformly homogeneous club_AD Forcing Axioms Subtle cardinal Absoluteness square Lipschitz reduction ZFC construction Generalized Clubs Diamond approachability ideal Hereditarily Lindelöf space Uniformly coherent Coherent tree Forcing Distributive tree ccc strongly bounded groups Slim tree Luzin set Strongly compact cardinal Knaster Foundations Closed coloring incompactness weak square Axiom R PFA Club Guessing Subadditive HOD Large Cardinals Constructible Universe Almost-disjoint family regressive Souslin tree Prikry-type forcing Singular Density Almost Souslin Commutative projection system Rado's conjecture 54G20 Uniformization square principles Precaliber Selective Ultrafilter Singular cofinality Strongly Luzin set Ascent Path Vanishing levels SNR Rock n' Roll specializable Souslin tree Diamond-sharp Erdos-Hajnal graphs countably metacompact diamond star Cardinal Invariants Hindman's Theorem OCA Commutative cancellative semigroups Mandelbrot set Fast club polarized partition relation Shelah's Strong Hypothesis projective Boolean algebra Parameterized proxy principle Almost countably chromatic Amenable C-sequence stationary reflection Prevalent singular cardinals positive partition relation Fat stationary set Poset Microscopic Approach full tree sap transformations Nonspecial tree Martin's Axiom Jonsson cardinal Dowker space Chromatic number Ramsey theory over partitions Filter reflection O-space coloring number Sierpinski's onto mapping principle Weakly compact cardinal Square-Brackets Partition Relations Successor of Singular Cardinal b-scale Was Ulam right? higher Baire space very good scale Knaster and friends Dushnik-Miller unbounded function Singular cardinals combinatorics Ostaszewski square Diamond for trees Local Club Condensation. Well-behaved magma Non-saturation super-Souslin tree Ineffable cardinal Ulam matrix GMA Universal Sequences S-Space Whitehead Problem L-space Cardinal function Partition Relations Strong coloring C-sequence Generalized descriptive set theory Aronszajn tree weak diamond P-Ideal Dichotomy Kurepa Hypothesis Reduced Power Analytic sets Greatly Mahlo Antichain Postprocessing function PFA(S)[S] xbox Sigma-Prikry stationary hitting Hedetniemi's conjecture indecomposable ultrafilter Souslin Tree Respecting tree free Boolean algebra Countryman line Open Access Small forcing Subnormal ideal nonmeager set tensor product graph Minimal Walks middle diamond Cohen real Chang's conjecture Erdos Cardinal Rainbow sets Subtle tree property Successor of Regular Cardinal Intersection model Fodor-type reflection reflection principles
Author Archives: Assaf Rinot
A new small Dowker space
Joint work with Roy Shalev and Stevo Todorcevic. Abstract. It is proved that if there exists a Luzin set, or if either the stick principle or $\diamondsuit(\mathfrak b)$ hold, then an instance of the guessing principle $\clubsuit_{AD}$ holds at the … Continue reading
Was Ulam right? II: Small width and general ideals
Joint work with Tanmay Inamdar. Abstract. We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam’s theorem and its extension … Continue reading
Posted in Partition Relations, Publications
Tagged 03E02, 03E35, 03E55, C-sequence, Open Access, Subnormal ideal, Ulam matrix, Was Ulam right?
1 Comment
MFO workshop in Set Theory, January 2022
I gave an invited talk at the Set Theory meeting in Obwerwolfach, January 2022. Talk Title: A dual of Juhasz’ question Abstract: Juhasz asked whether $\clubsuit$ implies the existence of a Souslin tree. Here we settle the dual problem of … Continue reading
Posted in Invited Talks
Tagged club_AD, Dowker space
Comments Off on MFO workshop in Set Theory, January 2022
Complicated colorings, revisited
Joint work with Jing Zhang. Abstract. In a paper from 1997, Shelah asked whether $Pr_1(\lambda^+,\lambda^+,\lambda^+,\lambda)$ holds for every inaccessible cardinal $\lambda$. Here, we prove that an affirmative answer follows from $\square(\lambda^+)$. Furthermore, we establish that for every pair $\chi<\kappa$ of … Continue reading
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
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
