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Uniformly homogeneous OCA Rainbow sets Small forcing Uniformization Open Access free Souslin tree Successor of Regular Cardinal indecomposable ultrafilter Selective Ultrafilter Strong coloring super-Souslin tree Filter reflection Rado's conjecture C-sequence Subtle cardinal Large Cardinals stationary hitting Hereditarily Lindelöf space ccc Diamond-sharp Lipschitz reduction Universal Sequences Prevalent singular cardinals Aronszajn tree Cardinal function b-scale Dushnik-Miller Microscopic Approach Diamond for trees sap Almost Souslin Cohen real Whitehead Problem Erdos Cardinal approachability ideal Hedetniemi's conjecture Well-behaved magma Subtle tree property Fodor-type reflection weak square coloring number PFA(S)[S] Ascent Path Nonspecial tree Iterated forcing S-Space weak diamond Dowker space Sigma-Prikry Almost countably chromatic P-Ideal Dichotomy Postprocessing function Erdos-Hajnal graphs HOD free Boolean algebra full tree Kurepa Hypothesis Commutative projection system Diamond Countryman line Reduced Power Shelah's Strong Hypothesis Generalized Clubs diamond star AIM forcing Cardinal Invariants Slim tree Antichain tensor product graph Non-saturation GMA Knaster SNR countably metacompact Souslin Tree Ostaszewski square Analytic sets Singular cofinality Subadditive very good scale Forcing Axioms stick nonmeager set Club Guessing Poset higher Baire space Sierpinski's onto mapping principle Generalized descriptive set theory square principles Ineffable cardinal Chang's conjecture Constructible Universe regressive Souslin tree Mandelbrot set Was Ulam right Knaster and friends Vanishing levels strongly bounded groups polarized partition relation club_AD Strongly Luzin set Rock n' Roll positive partition relation Square-Brackets Partition Relations Successor of Singular Cardinal middle diamond unbounded function Intersection model Axiom R Prikry-type forcing Sakurai's Bell inequality Uniformly coherent Luzin set Ulam matrix reflection principles Ramsey theory over partitions Greatly Mahlo Reflecting stationary set specializable Souslin tree incompactness Weakly compact cardinal Closed coloring L-space Parameterized proxy principle Hindman's Theorem ZFC construction PFA Partition Relations 54G20 Foundations Jonsson cardinal stationary reflection Absoluteness Amenable C-sequence Fat stationary set O-space square Singular Density Forcing Strongly compact cardinal Commutative cancellative semigroups Fast club Chromatic number Precaliber Respecting tree projective Boolean algebra Local Club Condensation. Martin's Axiom Minimal Walks weak Kurepa tree transformations Coherent tree Subnormal ideal xbox Singular cardinals combinatorics Almost-disjoint family Distributive tree
Category Archives: Partition Relations
Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines
Joint work with Tanmay Inamdar. Abstract. It is proved that if there is an $\aleph_2$-Aronszajn line, then there is one that does not contain an $\aleph_2$-Countryman line. This solves a problem of Moore and stands in a sharp contrast with … 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 $X$ of character less than $\mathfrak b$, if $X\rightarrow(\text{top }{\omega+1})^1_\omega$, then $X\rightarrow(\text{top }{\alpha})^1_\omega$ for all $\alpha<\omega_1$. In addition, … Continue reading
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
1 Comment
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
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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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
