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