An extension theorem for Steiner systems
In the first of the two articles under review (this item), the author proves the following extension theorem for Steiner systems with \(t= 2\): There are functions \(v_0(k)\) and \(\lambda_0(k, v)\) such that for any \(\text{S}(2, k- 1; v){\mathbf D}\) with \(v\geq v_0(k)\) and any \(\lambda\geq \lambda_0(k, v)\) satisfying the necessary arithmetic conditions there exists a 3-design \(\text{S}_\lambda(3, k; v+ 1)\) for which the derived design at a suitable point consists of \(\lambda\) copies of \({\mathbf D}\). The main interest of this theorem lies in the fact that it is a principal ingredient in the proof of the following major result contained in the author's second article (next item). Given a prime power \(q\) and any positive integer \(v\) satisfying the (necessary) conditions \(v- 1\equiv 0\text{ mod}(q- 1), v(v^2- 1)\equiv 0\text{ mod} (q^2- 1)\), there exists a Steiner system \(\text{S}(3, q+ 1, vq^n+ 1)\) for all sufficiently large \(n\) (i.e., \(n\geq n_1(q, v)\)).
scientific article; zbMATH DE number 778521