Preface

Williams, N. H.

doi:10.1016/s0049-237x(08)70663-3

Cited by 10 publications

(4 citation statements)

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“…3.2 from [19] does not provide us the value of d R (ω n ). We provide a proof for the following theorem in Appendix A. Theorem 3.3.…”

Section: Ordinal Ramsey Relationmentioning

confidence: 96%

“…Williams in his book [19,Thm. 7.2.7] proved the following result, which extends Ramsey's theorem to ordinals ω n where n ∈ N. .…”

Section: Ordinal Ramsey Relationmentioning

confidence: 99%

“…We call the least number that satisfies the first question the Ramsey degree of the ordinal α. Williams in [19,Thm. 7.2.7] showed that the Ramsey degree exists for any ordinal ω n where n ∈ N. Here we further provide a formula for computing the Ramsey degree of an arbitrary α < ω ω ; see Theorem 3.3 and Appendix A.…”

Section: Introductionmentioning

confidence: 99%

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A Polychromatic Ramsey Theory for Ordinals

Huschenbett

Liu

2013

Mathematical Foundations of Computer Science 2013

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The Ramsey degree of an ordinal α is the least number n such that any colouring of the edges of the complete graph on α using finitely many colours contains an n-chromatic clique of order type α. The Ramsey degree exists for any ordinal α < ω ω . We provide an explicit expression for computing the Ramsey degree given α. We further establish a version of this result for automatic structures. In this version the ordinal and the colouring are presentable by finite automata and the clique is additionally required to be regular. The corresponding automatic Ramsey degree turns out to be greater than the set theoretic Ramsey degree. Finally, we demonstrate that a version for computable structures fails.

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“…3.2 from [19] does not provide us the value of d R (ω n ). We provide a proof for the following theorem in Appendix A. Theorem 3.3.…”

Section: Ordinal Ramsey Relationmentioning

confidence: 96%

“…Williams in his book [19,Thm. 7.2.7] proved the following result, which extends Ramsey's theorem to ordinals ω n where n ∈ N. .…”

Section: Ordinal Ramsey Relationmentioning

confidence: 99%

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A Polychromatic Ramsey Theory for Ordinals

Huschenbett

Liu

2013

Mathematical Foundations of Computer Science 2013

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“…Delta families were studied in Erdos and Rado [4] and we refer the reader to this paper for details when needed. We refer the reader to Williams [5] for any further set theoretical background.…”

Section: Moreover the Supremum In The Definition Of Rs E (P And) Is Amentioning

confidence: 99%

Families of partial representing sets

Balanda

1985

J Aust Math Soc A

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Assume GCH. Let κ, μ, Σ be cardinals, with κ infinite. Let be a family consisting of λ pairwise almost disjoint subsets of Σ each of size κ, whose union is Σ. In this note it is shown that for each μ with 1 ≤ μ ≤min(λ, Σ), there is a “large” almost disjoint family of μ-sized subsets of Σ, each member of having non-empty intersection with at least μ members of the family .

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Set mappings of unrestricted order

Gibbon

1983

Bull. Austral. Math. Soc.

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A set mapping on a set S is a function f mapping S into the powerset of S such that x ∉ f(x) for each x in S. The set map f has order θ if θ is the least cardinal such that |f(x)| < θ for each x in S. A subset H of S is free for f if x ∉ f(y) for all x, y in H. In this paper we use classical results about set mappings of large order to investigate conditions which ensure a large free set for set mappings of unrestricted order.

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Preface

Cited by 10 publications

References 0 publications

A Polychromatic Ramsey Theory for Ordinals

A Polychromatic Ramsey Theory for Ordinals

Families of partial representing sets

Set mappings of unrestricted order

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