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On Quadruple Systems

Published online by Cambridge University Press:  20 November 2018

Haim Hanani
Haim Hanani*
Affiliation:
Technion, Israel Institute of Technology, Haifa
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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 that

1

is the number of elements of S(l, m, n) and

is 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).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 145 - 157
Copyright
Copyright © Canadian Mathematical Society 1960

References

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