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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
Vanstone, S. A. (1973), ‘The extendibility of (r,1) designs’, Proc. Third Southeastern Conference on Numerical Math. Utilitas Math. Inc., Winnipeg, 409–418.Google Scholar
Totten, J. (1976), ‘Embedding the complement of two lines in a finite projective plane’, J. Austral. Math. Soc. 22, 27–34.CrossRefGoogle Scholar