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