Almost disjoint sets

It is intuitive and easy to see that if A is a countable set and \{ A_\gamma \}_{\gamma \in \Gamma} , where A_\gamma \subset A , is a disjoint collection of subsets, then \Gamma is countable. (Although one has to be careful with saying “intuitive” and “easy to see” in set theory.) A natural question is, what happens if we allow the sets A_{\gamma} to have nonempty but finite intersections? This question was posed in Halmos’ classic problem book Problems for Mathematicians: Young and Old, and this short post is dedicated to a few solutions.

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