On Quadruple Systems

( n — h\ 11 m — h\ I-hi I \l-hj is the number of those elements of S (I, 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 5(2, 7, 43). There arises a problem of finding a necessary and sufficient condition for the existence of S (I, m, n)} or more precisely, of finding—for given values of / and m—all values of n for which S (I, rn, n) exists. Already in 1852 Steiner (6) (see also (4)) raised the following problem: (a) For what integer A is it possible to form triples, out of N given elements, in such a way that every pair of elements appears in exactly one triple? (b) Assuming (a) solved we require the further possibility of forming quadruples so that any three elements, not already forming a triple, should appear in exactly one quadruple and that no quadruple should contain a triple. Does this impose new conditions on the number N? (c) Assuming (a) and (b) solved, can we moreover form quintuples so that any four elements, neither forming a quadruple nor containing a triple,