The number of infinite substructures

Dugald Macpherson; Alan H. Mekler; Saharon Shelah

doi:10.1017/S0305004100069668

The number of infinite substructures

Dugald Macpherson
, Alan H. Mekler
, Saharon Shelah

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Given a relational structure M and a cardinal λ < ǀMǀ, let ϕλ denote the number of isomorphism types of substructures of M of size λ. It is shown that if μ < λ are cardinals, and ǀMǀ is sufficiently larger than λ, then ϕμ ≤ ϕλ. A description is also given for structures with few substructures of given infinite cardinality.

Original language	English
Pages (from-to)	193-209
Number of pages	17
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Volume	109
Issue number	1
DOIs	https://doi.org/10.1017/S0305004100069668
State	Published - 1991

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abstract = "Given a relational structure M and a cardinal λ < ǀMǀ, let ϕλ denote the number of isomorphism types of substructures of M of size λ. It is shown that if μ < λ are cardinals, and ǀMǀ is sufficiently larger than λ, then ϕμ ≤ ϕλ. A description is also given for structures with few substructures of given infinite cardinality.",

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AB - Given a relational structure M and a cardinal λ < ǀMǀ, let ϕλ denote the number of isomorphism types of substructures of M of size λ. It is shown that if μ < λ are cardinals, and ǀMǀ is sufficiently larger than λ, then ϕμ ≤ ϕλ. A description is also given for structures with few substructures of given infinite cardinality.

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The number of infinite substructures

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