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