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Prevalent singular cardinals Sierpinski's onto mapping principle Slim tree Minimal Walks Dowker space transformations Non-saturation Whitehead Problem Hereditarily Lindelöf space Chang's conjecture Local Club Condensation. Distributive tree free Souslin tree PFA(S)[S] projective Boolean algebra Aronszajn tree Foundations Fast club reflection principles Constructible Universe indecomposable ultrafilter Strongly Luzin set positive partition relation Vanishing levels Rock n' Roll ZFC construction full tree Almost countably chromatic Rado's conjecture Forcing Axioms GMA Postprocessing function club_AD C-sequence Diamond Was Ulam right Sigma-Prikry Almost-disjoint family L-space Souslin Tree Lipschitz reduction Precaliber Diamond for trees Universal Sequences Erdos Cardinal Singular cardinals combinatorics stick Open Access Fodor-type reflection Analytic sets Dushnik-Miller b-scale Knaster and friends xbox Cohen real Sakurai's Bell inequality Axiom R Forcing Cardinal function Martin's Axiom approachability ideal Shelah's Strong Hypothesis nonmeager set OCA Luzin set Small forcing countably metacompact square principles Fat stationary set Mandelbrot set Filter reflection Rainbow sets Parameterized proxy principle Iterated forcing unbounded function SNR Ulam matrix Erdos-Hajnal graphs Antichain Ascent Path Club Guessing Partition Relations Nonspecial tree Reduced Power Generalized descriptive set theory weak Kurepa tree free Boolean algebra Singular Density middle diamond weak square Absoluteness tensor product graph ccc diamond star incompactness Singular cofinality Well-behaved magma Square-Brackets Partition Relations stationary reflection Jonsson cardinal S-Space Hindman's Theorem Strong coloring super-Souslin tree Subadditive Ineffable cardinal AIM forcing Reflecting stationary set regressive Souslin tree Uniformly coherent Kurepa Hypothesis PFA Ramsey theory over partitions P-Ideal Dichotomy Large Cardinals Weakly compact cardinal weak diamond Knaster Successor of Singular Cardinal Uniformization Microscopic Approach Subtle tree property Hedetniemi's conjecture very good scale Poset 54G20 O-space HOD square Amenable C-sequence Selective Ultrafilter Successor of Regular Cardinal Coherent tree Closed coloring Uniformly homogeneous Greatly Mahlo specializable Souslin tree Chromatic number Cardinal Invariants Commutative cancellative semigroups polarized partition relation stationary hitting Subtle cardinal Almost Souslin Subnormal ideal Prikry-type forcing Ostaszewski square Diamond-sharp higher Baire space coloring number Generalized Clubs sap strongly bounded groups
Author Archives: Assaf Rinot
Gdańsk Logic Conference, May 2023
I gave an invited talk at the first Gdańsk Logic Conference, May 2023. Talk Title: Was Ulam right? Abstract: An Ulam matrix is one of the earliest gems of infinite combinatorics. Around the same time of its discovery, another Polish … Continue reading
A series of lectures on Club_AD, February–March 2023
As part of the Thematic Program on Set Theoretic Methods in Algebra, Dynamics and Geometry (Fields Institute, January–June, 2023), Spencer Unger and I delivered a Graduate Course on Set Theory, Algebra and Analysis. My part of the course was a … Continue reading
Winter School in Abstract Analysis, January 2023
I gave a 3-lecture tutorial at the Winter School in Abstract Analysis in Steken, January 2023. Title: Club guessing Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove … Continue reading
Sums of triples in Abelian groups
Joint work with Ido Feldman. Abstract. Motivated by a problem in additive Ramsey theory, we extend Todorcevic’s partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for … Continue reading
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
Posted in Preprints, Squares and Diamonds
Tagged Club Guessing
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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
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
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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
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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
