Embedding finite linear spaces in projective planes. II (Q579619)

A linear space \({\mathfrak S}\) is an incidence structure such that each line has at least two points and any two points are on exactly one line. In this paper only finite linear spaces with at least two lines are considered. The order of \({\mathfrak S}\) is the maximal number of lines through a point, minus one. It is well known that each linear space can be embedded into an infinite projective plane. The authors investigate the question which linear spaces of order n are embeddable into a projective plane of the same order n. They obtain the following major result. Let \(n+1-a\) be the minimal number of points on a line. If a is sufficiently small then \({\mathfrak S}\) is embeddable (into a projective plane of the same order). More precisely: If \(4n> 6a^ 4+ 9a^ 3+ 19a^ 2+ 8a,\) then \({\mathfrak S}\) is embeddable into a projective plane \({\mathfrak P}\) of the same order n, and \({\mathfrak P}\) is unique up to isomorphism. The difficult proof uses the first part of the paper [Ann. Discrete Math. 30, 39-56 (1986; Zbl 0586.51012)].