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A generalization of a theorem of Totten

Published online by Cambridge University Press:  09 April 2009

R. C. Mullin  and
S. A. Vanstone
R. C. Mullin
Affiliation:
Department of Combinatorics, University of Waterloo Ontario, Canada.
S. A. Vanstone
Affiliation:
Department of Combinatorics, University of Waterloo Ontario, Canada.
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Abstract

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An (r, 1) system is pair (V, F) where V is a v-set and F is a family of non-null subsets of V (b in number) which satisfy the following.

(i) Every pair of distinct members of V occur in precisely one member of F;

(ii) Every member of V occurs in precisely r members of F.

A pseudo parallel complement PPC(n, α) is an (n+1, 1) system with ν = n2 – αn and b ≦ n2 + n – α in which there are at least n – α blocks of size n. A pseudo intersection complement PIC (n, α) is an (n+1, 1) system with ν=n2 – αn + α – 1 b ≦ n2+n – α in which there are at least n – α+1 blocks of size n-1. It is shown that for α ≧ 4, every PIC(n, α) can be embedded in a PPC(n, α – 1) and that for n > (α4 + 2α3 + 2α2 + α)/2 every PPC(n, α) can be embedded in a projective plant of order n. The latter generalizes a result of Totten (who proves the result for α = 1).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 22 , Issue 4 , December 1976 , pp. 494 - 500
Copyright
Copyright © Australian Mathematical Society 1976

References

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Mullin, R. C. and Vanstone, S. A. (1976), ‘On regular pairwise balanced designs of order 6 and index 1’, Utilitas Math.Google Scholar
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Totten, J. (1976), ‘Embedding the complement of two lines in a finite projective plane’, J. Austral. Math. Soc. 22, 27–34.CrossRefGoogle Scholar