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