Semilinear spaces and their remarkable subsets (Q1919315)

A semilinear space is a pair \((S, {\mathcal L})\) of points and lines such that (1) \({\mathcal L}\) is a covering of \(S\); (2) every line has at least two points; (3) through any two points there is at most one line. This notion generalizes linear spaces. Examples of semilinear spaces are graphs whose lines have size two or ruled algebraic varieties with respect to their lines. The authors consider subsets of semilinear spaces, namely cliques or anticliques consisting of pairwise joinable or pairwise non-joinable points, blocking sets and ovoids. These terms may coincide; for example every ovoid in a semilinear space is a maximal anticlique and a blocking set with respect to the maximal subspaces (Theorem 4). Using these subsets several propositions about subspaces of semilinear spaces are proved and applications to graphs and Steiner systems are given. The proofs are mainly quite elementary.