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