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