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The number of infinite substructures

Published online by Cambridge University Press:  24 October 2008

Dugald Macpherson
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London El 4NS
Alan H. Mekler
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
Saharon Shelah
Institute of Mathematics, The Hebrew University, Jerusalem, Israel


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

Research Article
Copyright © Cambridge Philosophical Society 1991

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