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Ramsey theory over partitions Vanishing levels Hereditarily Lindelöf space projective Boolean algebra Non-saturation Lipschitz reduction strongly bounded groups Uniformization weak Kurepa tree approachability ideal P-Ideal Dichotomy regressive Souslin tree Antichain Distributive tree Uniformly homogeneous Sierpinski's onto mapping principle tensor product graph Axiom R Was Ulam right Poset Rainbow sets Iterated forcing C-sequence Absoluteness Singular cardinals combinatorics positive partition relation Chang's conjecture Selective Ultrafilter unbounded function stationary hitting coloring number diamond star weak square Open Access Rado's conjecture reflection principles Knaster and friends Amenable C-sequence PFA(S)[S] xbox incompactness Singular Density weak diamond Knaster Forcing Axioms AIM forcing Ineffable cardinal Minimal Walks Jonsson cardinal square principles Erdos Cardinal Souslin Tree HOD Sakurai's Bell inequality Partition Relations L-space club_AD Chromatic number Successor of Regular Cardinal Subadditive Ascent Path Subtle cardinal Fodor-type reflection Generalized Clubs Mandelbrot set Dowker space Aronszajn tree Fat stationary set Singular cofinality Rock n' Roll stick free Boolean algebra O-space Cohen real polarized partition relation Strongly Luzin set Luzin set Universal Sequences Diamond for trees Greatly Mahlo Reflecting stationary set Coherent tree b-scale Hindman's Theorem Filter reflection GMA Commutative cancellative semigroups Analytic sets ccc specializable Souslin tree Ostaszewski square higher Baire space Reduced Power Successor of Singular Cardinal Strong coloring square PFA indecomposable ultrafilter nonmeager set Hedetniemi's conjecture Uniformly coherent Dushnik-Miller Well-behaved magma Slim tree Cardinal function 54G20 Local Club Condensation. Almost countably chromatic Closed coloring countably metacompact Constructible Universe Fast club Subnormal ideal Nonspecial tree Precaliber Sigma-Prikry Martin's Axiom Subtle tree property Club Guessing Foundations transformations Parameterized proxy principle Erdos-Hajnal graphs ZFC construction Diamond-sharp full tree Weakly compact cardinal Almost Souslin Generalized descriptive set theory Shelah's Strong Hypothesis Small forcing middle diamond super-Souslin tree stationary reflection OCA very good scale Prevalent singular cardinals SNR Cardinal Invariants Prikry-type forcing Forcing Large Cardinals sap free Souslin tree Postprocessing function Ulam matrix Almost-disjoint family Square-Brackets Partition Relations Kurepa Hypothesis S-Space Whitehead Problem Microscopic Approach Diamond
Tag Archives: 03E35
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
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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
Fake Reflection
Joint work with Gabriel Fernandes and Miguel Moreno. Abstract. We introduce a generalization of stationary set reflection which we call filter reflection, and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We … Continue reading
Knaster and friends II: The C-sequence number
Joint work with Chris Lambie-Hanson. Abstract. Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable … Continue reading
A remark on Schimmerling’s question
Joint work with Ari Meir Brodsky. Abstract. Schimmerling asked whether $\square^*_\lambda$ together with GCH entails the existence of a $\lambda^+$-Souslin tree, for a singular cardinal $\lambda$. Here, we provide an affirmative answer under the additional assumption that there exists a … Continue reading
Weak square and stationary reflection
Joint work with Gunter Fuchs. Abstract. It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, stationary reflection, weak square
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A forcing axiom deciding the generalized Souslin Hypothesis
Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
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