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