Separability properties of almost-disjoint families of sets. (English) Zbl 0246.05002

Several results are proved on almost disjoint sets and many new problems are raised. Among others the authors prove the following conjecture of Hechler: Let \(F\) be a family of infinite sets \(\{A_\alpha \}\) satisfying \(|A_{\alpha_1} \cap A_{\alpha_2}| < \aleph_0\). Assume further that the family has chromatic number \(>2\) (or does not have property \(B\)) i.e. if a set has a non-empty intersection with every \(A\), then it contains at least one of them. Then there must be two \(A\)’s which are disjoint. Further in general it is not true that there are three \(A\)’s which are pairwise disjoint.


05A05 Permutations, words, matrices
03E05 Other combinatorial set theory
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