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 ν–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 ν=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 ν=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.