A Spernerk‐partition system on a set X is a set of partitions of X into k classes such that the classes of the partitions form a Sperner set system (so no class from a partition is a subset of a class from another partition). These systems were defined by Meagher, Moura, and Stevens in [6] who showed that if |X|=kℓ, then the largest Sperner k‐partition system has size 1k0pt|X|ℓ. In this paper, we find bounds on the size of the largest Sperner k‐partition system where k does not divide the size of X, specifically, we give upper and lower bounds when |X|=2k+1, |X|=2k+2 and |X|=3k−1.
Cover-free families (CFFs) were considered from different subjects by numerous researchers. In this article, we mainly consider explicit constructions of (2; d)-cover-free families. We also determine the size of optimal 2-cover-free-families on 9, 10, and 11 points. Related separating hash families, which can be used to construct CFFs, are also discussed.
Let m and t be positive integers with t 2. An (m, t)-splitting system is a pair (X, B) where |X| = m and B is a collection of subsets of X called blocks such that for every Y ⊆ X with |Y | = t, there exists a block B ∈ B such that |B ∩ Y | = t/2 . An (m, t)-splitting system is uniform if every block has size m/2 . In this paper, we give several constructions and bounds for splitting systems, concentrating mainly on the case t = 3. We consider uniform splitting systems as well as other splitting systems with special properties, including disjunct and regular splitting systems. Some of these systems have interesting connections with other types of set systems.
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