Mullin, R. C. ; Singhi, N. M. ; Vanstone, S. A. (1977) Embedding the affine complement of three intersecting lines in a finite projective plane Journal of the Australian Mathematical Society, Series A, 24 . pp. 458464. ISSN 03343316

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Related URL: http://dx.doi.org/10.1017/S1446788700020826
Abstract
An (r, 1)–design is a pair (V, F) where V is a ν–set and F is a family of nonnull subsets of V (b in number) which satisfy the following. (1) Every pair of distinct members of V is contained in precisely one member of F. (2) Every member of V occurs in precisely r members of F. A pseudo parallel complement PPC(n, α) is an (n+1, 1)–design with ν=n^{2}−αn and b≦n^{2}+n−α in which there are at least n−α a blocks of size n. A pseudo intersecting complement PIC(n, α) is an (n+1, 1)–design with ν=n^{2}−αn+α−1 and b≦n^{2}+n−α in which there are at least n−α+1 blocks of size n−1. It has previously been shown that for α≦4, every PIC(n, α) can be embedded in a PPC(n, α−1) and that for n>(α^{4}−2α^{3}+2α^{2}+α−2)/2, every PPC(n, α) can be embedded in a finite projective plane of order n. In this paper we investigate the case of α=3 and show that any PIC(n, 3) is embeddable in a PPC(n,2) provided n≧14.
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