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Diamond Entangled linear order Vanishing levels unbounded function Greatly Mahlo Small forcing Foundations Closed coloring Successor of Regular Cardinal Forcing Axioms perfectly normal sap Postprocessing function Local Club Condensation. 54G20 Was Ulam right? strongly bounded groups Well-behaved magma Distributive tree Ramsey theory over partitions Weakly compact cardinal higher Baire space stationary hitting PFA square weak Kurepa tree Erdos Cardinal Monotonically far SNR approachability ideal Uniformly coherent Microscopic Approach Square-Brackets Partition Relations Fat stationary set Sigma-Prikry Rainbow sets Subnormal ideal full tree Strongly Luzin set club_AD indecomposable filter Knaster Jonsson cardinal Mandelbrot set Nonspecial tree ZFC construction Fast club polarized partition relation Sakurai's Bell inequality P-Ideal Dichotomy Sierpinski's onto mapping principle Forcing Cardinal Invariants nonmeager set Diamond for trees Subadditive Almost Souslin Singular cofinality Partition Relations very good scale regressive Souslin tree Club Guessing Forcing with side conditions weak square Absoluteness O-space Prevalent singular cardinals projective Boolean algebra Aronszajn tree OCA Commutative projection system Iterated forcing Luzin set GMA S-Space stationary reflection Ulam matrix Antichain Minimal Walks Generalized descriptive set theory Ascending path Hedetniemi's conjecture Respecting tree stick Intersection model Prikry-type forcing Knaster and friends Analytic sets Whitehead Problem Chromatic number Cohen real Reflecting stationary set transformations free Boolean algebra Filter reflection tensor product graph Almost countably chromatic Singular Density weak diamond Rado's conjecture Chang's conjecture Dowker space Subtle tree property Kurepa Hypothesis Strongly compact cardinal incompactness Axiom R coloring number HOD Diamond-sharp Uniformly homogeneous Partition relations for trees Hindman's Theorem Parameterized proxy principle b-scale Universal Sequences L-space Martin's Axiom Dushnik-Miller Subtle cardinal Poset reflection principles Precaliber Non-saturation C-sequence specializable Souslin tree Coherent tree Singular cardinals combinatorics PFA(S)[S] Slim tree xbox Strong coloring diamond star Lipschitz reduction ccc free Souslin tree super-Souslin tree square principles Hereditarily Lindelöf space Almost-disjoint family middle diamond Selective Ultrafilter AIM forcing Large Cardinals Successor of Singular Cardinal Countryman line Rock n' Roll positive partition relation Ostaszewski square Cardinal function Constructible Universe Open Access Generalized Clubs Uniformization Reduced Power Interval topology on trees Ineffable cardinal Ascent Path countably metacompact Erdos-Hajnal graphs Souslin Tree Shelah's Strong Hypothesis Fodor-type reflection Amenable C-sequence Commutative cancellative semigroups
Category Archives: Partition Relations
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
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
Ramsey theory over partitions I: Positive Ramsey relations from forcing axioms
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, a correspondence between anti-Ramsey properties of partitions and chain conditions of the natural forcing notions … Continue reading
Posted in Partition Relations, Publications
Tagged 03E02, 03E17, 03E35, GMA, Martin's Axiom, positive partition relation, Ramsey theory over partitions
1 Comment
Transformations of the transfinite plane
Joint work with Jing Zhang. Abstract. We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals. To exemplify: we prove that for every … Continue reading
Knaster and friends I: Closed colorings and precalibers
Joint work with Chris Lambie-Hanson. Abstract. The productivity of the $\kappa$-chain condition, where $\kappa$ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of $\kappa$-cc posets whose squares … Continue reading
Strong failures of higher analogs of Hindman’s Theorem
Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading
Posted in Groups, Partition Relations, Publications
Tagged 03E02, 03E35, 03E75, 05A17, 05D10, 11P99, 20M14, Chang's conjecture, Commutative cancellative semigroups, Entangled linear order, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal, ZFC construction
1 Comment
Chain conditions of products, and weakly compact cardinals
Abstract. The history of productivity of the $\kappa$-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every … Continue reading
Posted in Partition Relations, Publications
Tagged Aronszajn tree, ccc, Fat stationary set, Minimal Walks, square, Weakly compact cardinal
2 Comments
Complicated colorings
Abstract. If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^\lambda_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. (Recall that $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ asserts the existence of a coloring $d:[\lambda]^2\rightarrow\lambda$ such that for any family $\mathcal A\subseteq[\lambda]^{<\kappa}$ of size $\lambda$, consisting of pairwise … Continue reading
Posted in Partition Relations, Publications
Tagged Minimal Walks, Open Access, Square-Brackets Partition Relations
2 Comments
Rectangular square-bracket operation for successor of regular cardinals
Joint work with Stevo Todorcevic. Extended Abstract: Consider the coloring statement $\lambda^+\nrightarrow[\lambda^+;\lambda^+]^2_{\lambda^+}$ for a given regular cardinal $\lambda$: In 1990, Shelah proved the above for $\lambda>2^{\aleph_0}$; In 1991, Shelah proved the above for $\lambda>\aleph_1$; In 1997, Shelah proved the above … Continue reading
