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full tree Hereditarily Lindelöf space positive partition relation Strongly Luzin set xbox Absoluteness free Souslin tree Well-behaved magma Fodor-type reflection Shelah's Strong Hypothesis SNR Almost-disjoint family Universal Sequences polarized partition relation Martin's Axiom ccc Minimal Walks Commutative cancellative semigroups Slim tree very good scale Ineffable cardinal Analytic sets Chang's conjecture Subtle cardinal stick Almost Souslin Aronszajn tree diamond star specializable Souslin tree ZFC construction PFA(S)[S] Was Ulam right Whitehead Problem club_AD sap Local Club Condensation. Erdos-Hajnal graphs Sierpinski's onto mapping principle projective Boolean algebra Antichain coloring number Large Cardinals Open Access 54G20 Subnormal ideal P-Ideal Dichotomy Amenable C-sequence indecomposable ultrafilter Kurepa Hypothesis Club Guessing stationary reflection Generalized Clubs Singular cofinality Hindman's Theorem Lipschitz reduction Ostaszewski square transformations Uniformization Strong coloring Precaliber HOD Cardinal function tensor product graph Diamond Cohen real Parameterized proxy principle Iterated forcing Erdos Cardinal L-space Souslin Tree Dowker space higher Baire space C-sequence Knaster and friends Subadditive Mandelbrot set PFA Fast club Small forcing Dushnik-Miller Closed coloring countably metacompact Singular Density Distributive tree Uniformly coherent approachability ideal Non-saturation Chromatic number incompactness Ramsey theory over partitions weak diamond strongly bounded groups Cardinal Invariants Filter reflection Reduced Power reflection principles Uniformly homogeneous Reflecting stationary set unbounded function Poset Greatly Mahlo square Postprocessing function Sigma-Prikry Ulam matrix Singular cardinals combinatorics Constructible Universe regressive Souslin tree Generalized descriptive set theory GMA Square-Brackets Partition Relations Ascent Path super-Souslin tree Prevalent singular cardinals weak square Weakly compact cardinal square principles stationary hitting Prikry-type forcing nonmeager set Axiom R middle diamond Knaster S-Space Foundations Subtle tree property Coherent tree Almost countably chromatic Sakurai's Bell inequality Jonsson cardinal Hedetniemi's conjecture Rock n' Roll Successor of Singular Cardinal O-space Luzin set Diamond-sharp free Boolean algebra Diamond for trees Fat stationary set Rainbow sets Successor of Regular Cardinal Forcing weak Kurepa tree Rado's conjecture AIM forcing Nonspecial tree Partition Relations b-scale Vanishing levels OCA Selective Ultrafilter Microscopic Approach Forcing Axioms
Category Archives: Partition Relations
Perspectives on Set Theory, November 2023
I gave an invited talk at the Perspectives on Set Theory conference, November 2023. Talk Title: May the successor of a singular cardinal be Jónsson? Abstract: We’ll survey what’s known about the question in the title and collect ten open … Continue reading
Posted in Invited Talks, Open Problems, Partition Relations
Tagged Jonsson cardinal, Successor of Singular Cardinal
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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, 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
Posted in Partition Relations, Publications
Tagged Ascent Path, Knaster and friends, Open Access, P-Ideal Dichotomy, sap, square, Subadditive, Uniformly coherent
1 Comment
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
2 Comments
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