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