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incompactness Lipschitz reduction full tree Weakly compact cardinal stick Luzin set Ramsey theory over partitions super-Souslin tree sap Dowker space Subtle cardinal stationary hitting Well-behaved magma Ascent Path Erdos-Hajnal graphs Poset Microscopic Approach Successor of Regular Cardinal Strongly Luzin set Strongly compact cardinal Foundations Club Guessing Hindman's Theorem Sakurai's Bell inequality very good scale Iterated forcing perfectly normal Closed coloring Almost countably chromatic Rainbow sets Respecting tree stationary reflection Selective Ultrafilter Souslin Tree Ascending path Strong coloring Parameterized proxy principle Chang's conjecture Hedetniemi's conjecture Postprocessing function coloring number HOD square Successor of Singular Cardinal S-Space Absoluteness Mandelbrot set specializable Souslin tree SNR Uniformly coherent Shelah's Strong Hypothesis Cohen real Diamond-sharp OCA Fodor-type reflection Large Cardinals Singular cardinals combinatorics Universal Sequences C-sequence Partition relations for trees nonmeager set Singular cofinality unbounded function Intersection model Jonsson cardinal Amenable C-sequence Cardinal function regressive Souslin tree Precaliber Analytic sets Was Ulam right? Knaster Greatly Mahlo Entangled linear order Non-saturation Erdos Cardinal 54G20 Ostaszewski square projective Boolean algebra Local Club Condensation. Nonspecial tree reflection principles diamond star Dushnik-Miller higher Baire space Square-Brackets Partition Relations Commutative projection system approachability ideal polarized partition relation Distributive tree Subnormal ideal Hereditarily Lindelöf space Whitehead Problem weak diamond Chromatic number Ineffable cardinal weak square Monotonically far weak Kurepa tree Singular Density P-Ideal Dichotomy Knaster and friends Kurepa Hypothesis free Boolean algebra Small forcing Martin's Axiom Reduced Power L-space Uniformization Reflecting stationary set countably metacompact Generalized descriptive set theory Sierpinski's onto mapping principle Commutative cancellative semigroups ccc Subtle tree property Almost-disjoint family Rock n' Roll GMA Forcing with side conditions Antichain square principles PFA tensor product graph Forcing Axioms O-space Countryman line Minimal Walks positive partition relation b-scale Aronszajn tree Prevalent singular cardinals Rado's conjecture strongly bounded groups free Souslin tree Cardinal Invariants Fat stationary set Prikry-type forcing xbox Filter reflection Generalized Clubs club_AD transformations Open Access Slim tree Fast club Sigma-Prikry Interval topology on trees PFA(S)[S] Vanishing levels Coherent tree Ulam matrix Axiom R Subadditive Diamond Partition Relations Uniformly homogeneous middle diamond Forcing Almost Souslin ZFC construction AIM forcing indecomposable filter Constructible Universe Diamond for trees
Tag Archives: Open Access
Sums of triples in Abelian groups
Joint work with Ido Feldman. Abstract. Motivated by a problem in additive Ramsey theory, we extend Todorcevic’s partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then for … Continue reading
A club guessing toolbox I
Joint work with Tanmay Inamdar. Abstract. Club guessing principles were introduced by Shelah as a weakening of Jensen’s diamond. Most spectacularly, they were used to prove Shelah’s ZFC bound on the power of the first singular cardinal. These principles have … Continue reading
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
Knaster and friends III: Subadditive colorings
Joint work with Chris Lambie-Hanson. Abstract. We continue our study of strongly unbounded colorings, this time focusing on subadditive maps. In Part I of this series, we showed that, for many pairs of infinite cardinals $\theta < \kappa$, the existence … Continue reading
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
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
Sigma-Prikry forcing I: The Axioms
Joint work with Alejandro Poveda and Dima Sinapova. Abstract. We introduce a class of notions of forcing which we call $\Sigma$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality … Continue reading
Higher Souslin trees and the GCH, revisited
Abstract. It is proved that for every uncountable cardinal $\lambda$, GCH+$\square(\lambda^+)$ entails the existence of a $\text{cf}(\lambda)$-complete $\lambda^+$-Souslin tree. In particular, if GCH holds and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in Godel’s constructible universe, improving … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, Open Access, regressive Souslin tree, Souslin Tree, square, Weakly compact cardinal, xbox
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Reduced powers of Souslin trees
Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed … Continue reading
Complicated colorings
Abstract. If $\lambda,\kappa$ are regular cardinals, $\lambda>\kappa^+$, and $E^\lambda_{\ge\kappa}$ admits a nonreflecting stationary set, then $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ holds. (Recall that $\text{Pr}_1(\lambda,\lambda,\lambda,\kappa)$ asserts the existence of a coloring $d:[\lambda]^2\rightarrow\lambda$ such that for any family $\mathcal A\subseteq[\lambda]^{<\kappa}$ of size $\lambda$, consisting of pairwise … Continue reading
Posted in Partition Relations, Publications
Tagged Minimal Walks, Open Access, Square-Brackets Partition Relations
2 Comments
