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Kurepa Hypothesis Coherent tree Diamond-sharp Was Ulam right P-Ideal Dichotomy Luzin set Almost Souslin Foundations SNR Prikry-type forcing strongly bounded groups ccc Absoluteness HOD Distributive tree Forcing Axioms Parameterized proxy principle Fat stationary set Subnormal ideal full tree Axiom R Hindman's Theorem Partition Relations Countryman line club_AD Non-saturation Rado's conjecture Commutative cancellative semigroups Strongly compact cardinal Strongly Luzin set tensor product graph b-scale OCA weak diamond unbounded function middle diamond incompactness coloring number Cardinal function Analytic sets C-sequence Generalized descriptive set theory Local Club Condensation. approachability ideal Whitehead Problem positive partition relation Ascent Path Reduced Power square principles Martin's Axiom Reflecting stationary set S-Space higher Baire space Uniformly homogeneous L-space stationary hitting PFA(S)[S] Postprocessing function free Boolean algebra Intersection model Cohen real weak Kurepa tree AIM forcing Sierpinski's onto mapping principle Fast club Hereditarily Lindelöf space Large Cardinals Chang's conjecture 54G20 xbox Closed coloring Sakurai's Bell inequality Knaster and friends transformations Almost-disjoint family Subadditive Hedetniemi's conjecture Precaliber Square-Brackets Partition Relations Souslin Tree Dushnik-Miller indecomposable ultrafilter PFA Iterated forcing Uniformization regressive Souslin tree Singular cardinals combinatorics Mandelbrot set Subtle cardinal weak square Successor of Regular Cardinal Erdos-Hajnal graphs Minimal Walks Universal Sequences countably metacompact Shelah's Strong Hypothesis Prevalent singular cardinals Aronszajn tree Commutative projection system Chromatic number polarized partition relation O-space Nonspecial tree Filter reflection Successor of Singular Cardinal Erdos Cardinal square Diamond Poset reflection principles sap Microscopic Approach stick Constructible Universe Club Guessing Ineffable cardinal Jonsson cardinal Singular cofinality projective Boolean algebra Rainbow sets Slim tree Amenable C-sequence Sigma-Prikry diamond star Respecting tree Weakly compact cardinal Selective Ultrafilter Knaster Rock n' Roll free Souslin tree Ostaszewski square very good scale Lipschitz reduction super-Souslin tree Dowker space Strong coloring GMA Ramsey theory over partitions Singular Density Subtle tree property Greatly Mahlo Well-behaved magma Almost countably chromatic Small forcing Vanishing levels stationary reflection Open Access Uniformly coherent Diamond for trees Fodor-type reflection ZFC construction Forcing Cardinal Invariants Generalized Clubs Antichain specializable Souslin tree Ulam matrix nonmeager set
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, 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, Preprints
Tagged 03E02, 03E75, 20A15, 20E15, 20F06, Jonsson cardinal, 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, Open Access, Subnormal ideal, Ulam matrix, Was Ulam right
1 Comment
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
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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, 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
