On simple partitions of $[κ]^{κ}$

David Asperó

Displaying similar documents to “On simple partitions of ${[κ]}^{κ}$ ”

Partition ideals below $ℵ_{ω}$

P. Dodos, J. Lopez-Abad, S. Todorcevic (2012)

Fundamenta Mathematicae

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Motivated by an application to the unconditional basic sequence problem appearing in our previous paper, we introduce analogues of the Laver ideal on ℵ₂ living on index sets of the form ${[ℵ_{k}]}^{ω}$ and use this to refine the well-known high-dimensional polarized partition relation for $ℵ_{ω}$ of Shelah.

Internally club and approachable for larger structures

John Krueger (2008)

Fundamenta Mathematicae

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We generalize the notion of a fat subset of a regular cardinal κ to a fat subset of $P_{κ} (X)$ , where κ ⊆ X. Suppose μ < κ, $μ^{< μ} = μ$ , and κ is supercompact. Then there is a generic extension in which κ = μ⁺⁺, and for all regular λ ≥ μ⁺⁺, there are stationarily many N in ${[H (λ)]}^{μ ⁺}$ which are internally club but not internally approachable.

Assertion Q distinguishes topologically ω* and $𝔪$ when $𝔪$ regular and $𝔪 > ω$

R. Frankiewicz (1978)

Colloquium Mathematicae

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Extension of point-finite partitions of unity

Haruto Ohta, Kaori Yamazaki (2006)

Fundamenta Mathematicae

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A subspace A of a topological space X is said to be $P^{γ}$ -embedded ( $P^{γ}$ (point-finite)-embedded) in X if every (point-finite) partition of unity α on A with |α| ≤ γ extends to a (point-finite) partition of unity on X. The main results are: (Theorem A) A subspace A of X is $P^{γ}$ (point-finite)-embedded in X iff it is $P^{γ}$ -embedded and every countable intersection B of cozero-sets in X with B ∩ A = ∅ can be separated from A by a cozero-set in X. (Theorem B) The product A × [0,1] is $P^{γ}$ (point-finite)-embedded...

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

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Given a bounded open set $Ω$ in $ℝ^{n}$ (or in a Riemannian manifold) and a partition of $Ω$ by $k$ open sets $D_{j}$ , we consider the quantity ${𝚖𝚊𝚡}_{j} λ (D_{j})$ where $λ (D_{j})$ is the ground state energy of the Dirichlet realization of the Laplacian in $D_{j}$ . If we denote by $ℒ_{k} (Ω)$ the infimum over all the $k$ -partitions of ${𝚖𝚊𝚡}_{j} λ (D_{j})$ , a minimal $k$ -partition is then a partition which realizes the infimum. When $k = 2$ , we find the two nodal domains of a second eigenfunction, but the analysis of higher $k$ ’s is non trivial and quite interesting. In this...

On starshapedness of the union of closed sets in $R^{n}$

Krzysztof Kołodziejczyk (1987)

Colloquium Mathematicae

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Classifying homogeneous ultrametric spaces up to coarse equivalence

Taras Banakh, Dušan Repovš (2016)

Colloquium Mathematicae

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For every metric space X we introduce two cardinal characteristics $c o v^{♭} (X)$ and $c o v^{♯} (X)$ describing the capacity of balls in X. We prove that these cardinal characteristics are invariant under coarse equivalence, and that two ultrametric spaces X,Y are coarsely equivalent if $c o v^{♭} (X) = c o v^{♯} (X) = c o v^{♭} (Y) = c o v^{♯} (Y)$ . This implies that an ultrametric space X is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if $c o v^{♭} (X) = c o v^{♯} (X)$ . Moreover, two isometrically homogeneous ultrametric spaces X,Y are coarsely equivalent if and...

The existence of $P (α)$ -points of N* for $ℵ_{0} α < 𝔠$

A. Szymański (1977)

Colloquium Mathematicae

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Indestructible colourings and rainbow Ramsey theorems

Lajos Soukup (2009)

Fundamenta Mathematicae

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We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c. It is also consistent that $2^{ω ₁}$ is arbitrarily large,...