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