## 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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