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Chang's conjecture Antichain Singular cardinals combinatorics Large Cardinals Subtle cardinal Diamond Vanishing levels reflection principles Strong coloring Subtle tree property sap Reduced Power GMA Selective Ultrafilter Cohen real Subadditive Successor of Singular Cardinal Prikry-type forcing Forcing approachability ideal Parameterized proxy principle Microscopic Approach Precaliber Analytic sets Entangled linear order Ulam matrix Ascent Path L-space PFA Minimal Walks higher Baire space Luzin set Postprocessing function diamond star Fat stationary set Forcing with side conditions Square-Brackets Partition Relations AIM forcing Absoluteness S-Space Martin's Axiom weak square Almost Souslin Shelah's Strong Hypothesis Sierpinski's onto mapping principle Ramsey theory over partitions stick Singular Density perfectly normal O-space Jonsson cardinal regressive Souslin tree Erdos-Hajnal graphs Weakly compact cardinal xbox Interval topology on trees Cardinal function Hedetniemi's conjecture Open Access Chromatic number Fodor-type reflection Universal Sequences full tree Partition Relations Singular cofinality middle diamond square principles Generalized Clubs Subnormal ideal Reflecting stationary set Uniformly homogeneous Knaster Ostaszewski square SNR Generalized descriptive set theory Sigma-Prikry specializable Souslin tree Ascending path PFA(S)[S] polarized partition relation Hindman's Theorem countably metacompact projective Boolean algebra Diamond-sharp Commutative projection system Intersection model Nonspecial tree Almost countably chromatic coloring number P-Ideal Dichotomy free Souslin tree transformations stationary hitting tensor product graph Iterated forcing Strongly compact cardinal b-scale Diamond for trees HOD Cardinal Invariants Whitehead Problem very good scale Was Ulam right? Uniformization Souslin Tree unbounded function Filter reflection Fast club weak Kurepa tree Monotonically far Small forcing C-sequence Strongly Luzin set Forcing Axioms Prevalent singular cardinals Aronszajn tree Kurepa Hypothesis 54G20 Closed coloring incompactness ZFC construction Constructible Universe indecomposable filter positive partition relation Amenable C-sequence Local Club Condensation. Poset Axiom R Coherent tree weak diamond Almost-disjoint family Knaster and friends Well-behaved magma Lipschitz reduction OCA Successor of Regular Cardinal ccc Hereditarily Lindelöf space Uniformly coherent Foundations Rock n' Roll club_AD Dowker space Sakurai's Bell inequality Countryman line super-Souslin tree Respecting tree free Boolean algebra Dushnik-Miller Commutative cancellative semigroups Erdos Cardinal Mandelbrot set Rado's conjecture Club Guessing strongly bounded groups Slim tree Non-saturation Greatly Mahlo Distributive tree square Partition relations for trees nonmeager set Rainbow sets Ineffable cardinal stationary reflection
Tag Archives: 03E55
Ketonen’s question and other cardinal sins
Joint work with Zhixing You and Jiachen Yuan. Abstract. Answering a question of Ketonen from the late 1970’s, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing … 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?
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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
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
Openly generated Boolean algebras and the Fodor-type reflection principle
Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading
Posted in Compactness, Publications
Tagged 03E35, 03E55, 03E65, 03E75, 03G05, 06E05, Axiom R, Fodor-type reflection, free Boolean algebra, projective Boolean algebra, Shelah's Strong Hypothesis, stationary reflection
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On the consistency strength of the Milner-Sauer conjecture
Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The conjecture is consistent and known … Continue reading
