(m, 3)-Splitting systems

Abstract

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 Ç 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] and an (m, t)-splitting system is disjunct or sperner if no block is a subset of another block. In this paper, we give several constructions and bounds for splitting systems, when t = 3. We consider uniform splitting systems as well as disjunct splitting systems. There tire many connections with other types of set systems.