Finite \(\{\) 1,3\(\}\)-semiaffine planes (Q1065385)

A \(\{\) 1,3\(\}\)-semiaffine plane is a linear space such that through any point not on a line \(\ell\) there is exactly one or three lines parallel to \(\ell\); the plane is a proper \(\{\) 1,3\(\}\)-semiaffine plane if there are instances of one parallel and also of three parallels. Let \(n+1\) be the maximum number of lines through a point in a finite \(\{\) 1,3\(\}\)- semiaffine plane S; then every line is either an n-line (a line containing just n points) or an (n-2)-line. If there is a point with fewer than \(n+1\) lines through it, then \(n=4\) and S is a particular linear space with just six points. If there are \(n+1\) lines through each point, then the number a of (n-2)-lines through a point is independent of the point, and so is the number b of n-lines through a point. The authors go on to prove the following result in this case. (i) If \(a=1\), then S is the complement of a Baer subplane in a projective plane of order 4. (ii) If \(b=1\) and \(n>23\), then S is the complement of three concurrent lines in a projective plane of order n. (iii) If \(a>1\) and \(b>1\), then either S is the complement of an \((n+2)\)-arc and one of its exterior lines in a projective plane of order n, or \(n=4\) and S is the complement of a unital in a projective plane of order 4.