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