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