On Quadruple Systems

Abstract

Given a set E of n elements we denote by S(l, m, n), (l ≤ m ≤ n) a system of subsets of E, having m elements each, such that every subset of E having l elements is contained in exactly one set of the system S (l, m, n).It is clear (3), that a necessary condition for the existence of S (l, m, n) is that1is the number of elements of S(l, m, n) andis the number of those elements of S (l, m, n) which contain h fixed elements of E.It is known that condition (1) is not sufficient for S(l, m, n) to exist. It has been proved that no finite projective geometry exists with 7 points on every line. This implies non-existence of S(2, 7, 43).