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On Block-Schematic Steiner Systems: S(t, t + 1, v)

Published online by Cambridge University Press:  20 November 2018

Mitsuo Yoshizawa
Mitsuo Yoshizawa*
Affiliation:
Keio University, Yokohama, Japan
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A Steiner system S(t, k, v) is a collection of k-subsets, called blocks, of a v-set of points with the property that any t-subset of points is contained in a unique block. We assume 1 < t < k < v. A Steiner system is called block-schematic if the blocks form an association scheme with the relations determined by size of intersection. Ito and Patton [3] proved that if S(4, 5, v) is block-schematic, then v = 11. The purpose of this paper is to extend this result, and we prove the following theorem.

THEOREM. A Steiner system S(t, t + 1, v) is block-schematic if and only if one of the following holds: (i) t = 2, (ii) t = 3, v = 8, (iii) t = 4, v = 11, (iv) t = 5, v = 12.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 6 , 01 December 1981 , pp. 1432 - 1438
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Bose, R. C., Strongly regular graphs, partial geometries, and partially balanced designs, Pacific J. Math. 13 (1963), 389–419.Google Scholar
2. Cameron, P. J., Near-regularity conditions for designs, Geometriae Dedicata 2 (1973), 213–223.Google Scholar
3. Ito, N. and Patton, W. H., On a class of Steiner 4-systems, unpublished.Google Scholar
4. Mendelsohn, N. S., A theorem on Steiner systems, Can. J. Math. 22 (1970), 1010–1015.Google Scholar
5. Mendelsohn, N.S. and Hung, S. H. Y., On the Steiner systems 5(3, 4, 14) and 5(4, 5, 15), Utilitas Math. 1 (1972), 5–95.Google Scholar
6. Witt, E., Ueber Steinersche Système, Abh. Hamburg 12 (1938), 265–275.Google Scholar