On near dimension in Steiner systems with parallelism and matroids (Q909662)

Let S be a 3-(v,q,1) Steiner system with an equivalence relation \(\|\) defined on the blocks of S such that for any block B and any two distinct points x and y, there is a unique block \(B'\) containing x and y with \(B'\| B\). The author introduces two ``near-dimensions'' \(\tilde d\) and \(d_ m\) on S. Suppose U is a subsystem of S (which also preserves \(\|)\). Then \(\tilde d(U)=n\) if \(n+1\) is the cardinality of an independent set (basis) that generates U and \(d_ m(U)=k\) if k is the least upper bound of ``m-chains'' \(U\supset U^ 1\supset...\supset U^ k=\emptyset\) where \(U^ i\) is a maximal subspace of \(U^{i-1}\). The main theorem is \(\tilde d(S)\)\(\leq d_ m(S)-1\). The author also constructs examples of the Steiner systems under consideration from a 2- transitive permutation group G with \(G_{xy}\) different from the identity group. It is further shown that particular types of Steiner systems of the above kind give rise to matroids.