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sap Strong coloring Martin's Axiom Hindman's Theorem strongly bounded groups Precaliber Was Ulam right? Successor of Singular Cardinal PFA Partition Relations Knaster and friends Strongly Luzin set Dowker space Singular cardinals combinatorics Square-Brackets Partition Relations Antichain Vanishing levels stationary reflection Greatly Mahlo Uniformly homogeneous C-sequence Closed coloring Distributive tree Singular cofinality Ascent Path Cardinal Invariants Club Guessing free Boolean algebra Aronszajn tree Chromatic number reflection principles Successor of Regular Cardinal Ramsey theory over partitions Ostaszewski square square b-scale Commutative cancellative semigroups Ulam matrix Almost Souslin Small forcing ccc Rock n' Roll Coherent tree regressive Souslin tree Chang's conjecture Well-behaved magma Shelah's Strong Hypothesis super-Souslin tree Open Access Generalized Clubs Axiom R Reduced Power unbounded function Mandelbrot set weak square middle diamond Intersection model specializable Souslin tree Large Cardinals OCA Fast club PFA(S)[S] Sakurai's Bell inequality countably metacompact O-space Prikry-type forcing Luzin set Erdos-Hajnal graphs Prevalent singular cardinals Rado's conjecture S-Space Subtle cardinal Selective Ultrafilter Almost-disjoint family Non-saturation stationary hitting approachability ideal Hereditarily Lindelöf space free Souslin tree Rainbow sets Respecting tree diamond star Foundations Weakly compact cardinal higher Baire space Hedetniemi's conjecture L-space Slim tree Subnormal ideal positive partition relation Souslin Tree Minimal Walks coloring number Nonspecial tree club_AD Fat stationary set Sigma-Prikry Whitehead Problem Universal Sequences P-Ideal Dichotomy Cohen real Amenable C-sequence Generalized descriptive set theory xbox Microscopic Approach square principles Commutative projection system Postprocessing function ZFC construction Forcing Axioms Uniformly coherent incompactness Singular Density Lipschitz reduction HOD Cardinal function transformations polarized partition relation Strongly compact cardinal very good scale Iterated forcing Absoluteness Fodor-type reflection projective Boolean algebra Diamond for trees AIM forcing SNR weak diamond Parameterized proxy principle Subadditive Subtle tree property Dushnik-Miller GMA Erdos Cardinal Constructible Universe Local Club Condensation. nonmeager set tensor product graph Countryman line Diamond 54G20 stick Filter reflection Almost countably chromatic full tree weak Kurepa tree Diamond-sharp Poset Reflecting stationary set Forcing Kurepa Hypothesis Uniformization Sierpinski's onto mapping principle Ineffable cardinal Knaster Analytic sets Jonsson cardinal indecomposable ultrafilter
Category Archives: Partition Relations
Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines
Joint work with Tanmay Inamdar. Abstract. We investigate global structural properties of linear orders of a fixed infinite size. It is classical that the countable linear orders and the continuum-sized orders exhibit contrasting behaviours. Modern results show that strong extensions … Continue reading
A counterexample related to a theorem of Komjáth and Weiss
Joint work with Rodrigo Rey Carvalho. Abstract. In a paper from 1987, Komjath and Weiss proved that for every regular topological space
Posted in Partition Relations, Preprints, Topology
Tagged 03E02, 54G20, Prikry-type forcing, ZFC construction
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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
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, Open Access, Subnormal ideal, Ulam matrix, Was Ulam right?
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Complicated colorings, revisited
Joint work with Jing Zhang. Abstract. In a paper from 1997, Shelah asked whether
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
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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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
