Infinite linear spaces
While counting is the main tool in the theory of finite linear spaces in the infinite case free constructions and transfinite induction are used. Some well-known results are generalized in the paper under review: There exist free incidence structures of any infinite cardinality with the property that any \(t\) points lie in exactly \(\lambda\) blocks and any \(s>\lambda\) blocks intersect in exactly \(\mu<t\) points. The same holds for \(s \leq \lambda\), \(t \leq \mu\). Let \(X\) be an infinite linear space with \(\alpha\) lines in which each line has \(\alpha\) points. Let \(\mu\) be a cardinal number satisfying \(1<\mu < \alpha\). Then there exists \(S \subset X\) meeting every line in \(\mu\) points. Furthermore, among other things it is shown that there exists a countable infinite Steiner triple system \(S\) with 2-transitive automorphism group such that any finite triple system occurs as a subsystem of \(S\). Several problems are posed in connection with the theorems.
scientific article; zbMATH DE number 599962
scientific article