A characterization of separating distinguished collections containing the maximum number of sets

AbstractPeleg [3] proved that g(n) = max{2(n − 1), [n24]} is an upper bound for the number of sets in collectionssatisfying certain strong separation conditions, and is a maximum under weaker conditions.In the present paper we find all the distinguished collections (Definition 2.2) of g(n) sets for n < 7, and prove that they all satisfy Peleg's weak conditions while none satisfies the strong conditions. For n > 7, we prove that there is a unique separating distinguished collection of g(n) sets satisfying the weak conditions, and that this collection also satisfies the strong conditions. We then prove that, for n ⩾ 4, the maximum number of sets in a collection satisfying the strong separation conditions is [n34]