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