Embedding the affine complement of three intersecting lines in a finite projective plane

Abstract

An (r, 1)–design is a 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. (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 v = n² – αn and b ≦ n² + 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 v = n² − αn + α − 1 and b ≦ n² + 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 > (α⁴ − 2α³ + 2α² + 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.