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