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Almost disjoint families: An application to linear algebra. (English) Zbl 0948.03054

The question central to this paper is the following. For a given infinite cardinal number \(\kappa\), let \(V\) be a \(\kappa\)-dimensional vector space over a countable field. What cardinalities are possible for a family \(A\) of subspaces of \(V\) which is maximal with respect to the property that whenever \(U\) and \(W\) are distinct members of \(A\) then \(U\cap W\) has dimension less than \(\kappa\)? It turns out that the axioms of ordinary set theory (ZFC) are insufficient to provide answers.
The corresponding question in set theory, given the infinite cardinal \(\kappa\) and a set \(S\) of size \(\kappa\), is to find the cardinality of a family \(A\) of \(\kappa\)-size subsets of \(S\) which is maximal with respect to the property that whenever \(a\) and \(b\) are distinct members of \(A\) then \(|a\cap b|<\kappa\). Such a family \(A\) is called a maximal almost disjoint family of sets, and determining the cardinalities possible for such families is a well-studied problem.
The author shows how the results and methods from maximal almost disjoint families of sets can be applied to families of subsets of a fixed basis of the vector space. The outcome is that the answers to the vector space problem parallel those from maximal almost disjoint families of sets.

MSC:

03E75 Applications of set theory
15A03 Vector spaces, linear dependence, rank, lineability
03E35 Consistency and independence results
03E05 Other combinatorial set theory
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