On maximal partial spreads in \(PG\)(\(n\), \(q\))
The paper under review contains some constructions of maximal partial spreads in \(PG(n, q)\). A partial spread in \(PG(n, q)\) is a set of pairwise disjoint lines. It is called maximal if it is not contained properly in a larger partial spread. In Section 2 of the paper, using a result on bipartite graphs (Lemma 2.1), the authors obtain three important lemmas for their constructions of a maximal partial spreads in \(PG(n, q)\). Then, they consider the cases \(n = 3\), \(n\geq 5\) odd, and \(n\geq 6\) even. In each case they give a construction of a maximal partial spread. When \(n = 3\), as the authors observe, their Construction 3.1 is almost identical with the one of \textit{A. Beutelspacher} [Geom. Dedicata 9, 425--449 (1980; Zbl 0377.50007)], but it enables the authors to construct maximal partial spreads smaller than the spreads constructed by Beutelspacher. Finally, in the cases \(n\geq 5\), they construct maximal partial spread in \(PG(n, q)\) of each size between \(c n q^{n - 2}\log q\) and \(c' q^{n - 1}\). Reviewer's remark. In Sections 2 and 5 some misprints occur. For example, sometimes they refer to Lemma 1.3 and to Theorem 1.3, but in the paper there is no lemma or theorem called 1.3.
scientific article; zbMATH DE number 1968861
scientific article