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Keywords
OCA Sigma-Prikry Uniformly homogeneous Mandelbrot set Countryman line square Shelah's Strong Hypothesis PFA Cohen real Successor of Regular Cardinal Square-Brackets Partition Relations HOD incompactness Rock n' Roll sap Precaliber Antichain Distributive tree Coherent tree P-Ideal Dichotomy Hereditarily Lindelöf space Constructible Universe Slim tree square principles Knaster and friends Commutative projection system polarized partition relation Chromatic number nonmeager set Forcing with side conditions Forcing stick Closed coloring Rainbow sets xbox Singular cofinality stationary reflection Respecting tree Diamond-sharp Fodor-type reflection Dushnik-Miller strongly bounded groups Cardinal Invariants weak square Weakly compact cardinal projective Boolean algebra diamond star Vanishing levels Was Ulam right? Partition relations for trees Reduced Power Selective Ultrafilter approachability ideal Well-behaved magma indecomposable filter transformations Reflecting stationary set Club Guessing free Boolean algebra Small forcing regressive Souslin tree Diamond for trees middle diamond ZFC construction Ramsey theory over partitions Ostaszewski square full tree Strongly Luzin set Interval topology on trees weak Kurepa tree L-space Hedetniemi's conjecture C-sequence Generalized Clubs perfectly normal super-Souslin tree Martin's Axiom stationary hitting Sakurai's Bell inequality Erdos-Hajnal graphs Uniformly coherent b-scale Jonsson cardinal Knaster coloring number S-Space Greatly Mahlo Filter reflection Postprocessing function Ulam matrix Open Access Intersection model free Souslin tree reflection principles O-space Fat stationary set specializable Souslin tree Subnormal ideal Generalized descriptive set theory Ascending path ccc Lipschitz reduction Parameterized proxy principle Commutative cancellative semigroups Fast club Absoluteness Subtle tree property Large Cardinals PFA(S)[S] Successor of Singular Cardinal Aronszajn tree Sierpinski's onto mapping principle Dowker space Amenable C-sequence Partition Relations unbounded function Entangled linear order tensor product graph Poset Souslin Tree Luzin set Non-saturation higher Baire space 54G20 Microscopic Approach Prikry-type forcing Cardinal function Strong coloring positive partition relation Local Club Condensation. Singular Density Ascent Path Almost-disjoint family Prevalent singular cardinals Chang's conjecture club_AD Strongly compact cardinal Iterated forcing weak diamond Kurepa Hypothesis Axiom R Ineffable cardinal Analytic sets Singular cardinals combinatorics Foundations Minimal Walks GMA Nonspecial tree Whitehead Problem very good scale Rado's conjecture AIM forcing Erdos Cardinal Hindman's Theorem Universal Sequences Subtle cardinal Monotonically far Almost Souslin countably metacompact Uniformization Subadditive SNR Forcing Axioms Almost countably chromatic Diamond
Tag Archives: 03E02
A counterexample related to a theorem of Komjáth and Weiss
Joint work with Rodrigo Rey Carvalho. Abstract. In a paper from 1987, Komjath and Weiss proved that for every regular topological space $X$ of character less than $\mathfrak b$, if $X\rightarrow(\text{top }{\omega+1})^1_\omega$, then $X\rightarrow(\text{top }{\alpha})^1_\omega$ for all $\alpha<\omega_1$. In addition, … Continue reading
Posted in Partition Relations, Preprints, Topology
Tagged 03E02, 54G20, Open Access, Prikry-type forcing, ZFC construction
Comments Off on A counterexample related to a theorem of Komjáth and Weiss
A Shelah group in ZFC
Joint work with Márk Poór. Abstract. In a paper from 1980, Shelah constructed an uncountable group all of whose proper subgroups are countable. Assuming the continuum hypothesis, he constructed an uncountable group $G$ that moreover admits an integer $n$ satisfying … Continue reading
Posted in Groups, Publications
Tagged 03E02, 03E75, 20A15, 20E15, 20F06, Jonsson cardinal, Open Access, Strong coloring, strongly bounded groups, Subadditive, ZFC construction
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Was Ulam right? II: Small width and general ideals
Joint work with Tanmay Inamdar. Abstract. We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam’s theorem and its extension … Continue reading
Posted in Partition Relations, Publications
Tagged 03E02, 03E35, 03E55, C-sequence, Kurepa Hypothesis, Open Access, Subnormal ideal, Ulam matrix, Was Ulam right?
2 Comments
Was Ulam right? I: Basic theory and subnormal ideals
Joint work with Tanmay Inamdar. Abstract. We introduce various coloring principles which generalize the so-called onto mapping principle of Sierpinski to larger cardinals and general ideals. We prove that these principles capture the notion of an Ulam matrix and allow … Continue reading
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
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
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
