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Generalized Clubs Generalized descriptive set theory Commutative cancellative semigroups C-sequence Cohen real Nonspecial tree Well-behaved magma nonmeager set Iterated forcing Subtle cardinal Singular cofinality Shelah's Strong Hypothesis Strongly Luzin set Subtle tree property Reflecting stationary set Prikry-type forcing square Non-saturation Strong coloring PFA(S)[S] Minimal Walks Uniformly homogeneous Strongly compact cardinal Interval topology on trees GMA Weakly compact cardinal regressive Souslin tree polarized partition relation SNR OCA approachability ideal unbounded function Uniformization Partition relations for trees Forcing with side conditions Slim tree Microscopic Approach P-Ideal Dichotomy Martin's Axiom Almost-disjoint family full tree coloring number Small forcing very good scale specializable Souslin tree Club Guessing Fat stationary set perfectly normal Universal Sequences xbox club_AD S-Space Ascending path stick Constructible Universe Vanishing levels countably metacompact Fodor-type reflection Intersection model Souslin Tree Entangled linear order Sierpinski's onto mapping principle Subadditive Sigma-Prikry strongly bounded groups Singular Density Forcing tensor product graph Prevalent singular cardinals Mandelbrot set incompactness Reduced Power ZFC construction Erdos-Hajnal graphs Poset Parameterized proxy principle free Boolean algebra Hereditarily Lindelöf space Hedetniemi's conjecture Dushnik-Miller middle diamond AIM forcing Commutative projection system Subnormal ideal stationary hitting Erdos Cardinal Amenable C-sequence Diamond Analytic sets Lipschitz reduction free Souslin tree Cardinal function Absoluteness Closed coloring Uniformly coherent Whitehead Problem Open Access L-space Almost countably chromatic Monotonically far Aronszajn tree Ostaszewski square Diamond-sharp 54G20 Rainbow sets Distributive tree sap Dowker space diamond star weak square Kurepa Hypothesis Greatly Mahlo Ramsey theory over partitions Luzin set higher Baire space PFA Selective Ultrafilter Forcing Axioms Singular cardinals combinatorics weak diamond Antichain Cardinal Invariants Ineffable cardinal Knaster and friends Local Club Condensation. Postprocessing function O-space Rado's conjecture b-scale Was Ulam right? Ascent Path Respecting tree HOD Large Cardinals ccc weak Kurepa tree projective Boolean algebra Countryman line positive partition relation Fast club Knaster indecomposable filter Successor of Singular Cardinal Ulam matrix Successor of Regular Cardinal Square-Brackets Partition Relations Precaliber Chang's conjecture Hindman's Theorem Sakurai's Bell inequality Coherent tree transformations Partition Relations Chromatic number Jonsson cardinal stationary reflection super-Souslin tree Rock n' Roll Diamond for trees square principles Almost Souslin reflection principles Filter reflection Foundations Axiom R
Category Archives: Partition Relations
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
Transforming rectangles into squares, with applications to strong colorings
Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading
