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