Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-15T06:49:38.027Z Has data issue: false hasContentIssue false
  • English
  • Français

On Quadruple Systems

Published online by Cambridge University Press:  20 November 2018

Haim Hanani
Haim Hanani*
Affiliation:
Technion, Israel Institute of Technology, Haifa
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Carmichael, R.D., Introduction to the theory of groups of finite order (New York, 1956). 415–441.Google Scholar
2. Moore, E.H., Concerning triple systems, Math. Annal. 43 (1893), 271–285.Google Scholar
3. Moore, E.H., Tactical memoranda, Amer. J. Math. 18 (1896), 264–303.Google Scholar
4. Netto, E., Lehrbuch der Combinatorik, zweite Auflage (Leipzig, 1927), pp. 202-220, 321–329.Google Scholar
5. Reiss, M., Ueber eine Steinersche combinatorische Aufgabe, J. reine und angew. Math., 56 (1859), 326–344.Google Scholar
6. Steiner, J., Combinatorische Aufgabe, J. reine und angew. Math. (1853), 181-182; also, Gesammelte Werke II (Berlin, 1884). pp. 435–436.Google Scholar
7. Witt, E., Ueber Steinersche Système, Abh. Math. Sem. Hamburg, 12 (1938), 265–275.Google Scholar
8. Tarry, G., Le problème des 36 officiers, C. R. Assoc. Franc. Av. Sci., 1 (1900), 122–123. 2 (1901), 170-203.Google Scholar