Minimal Enclosings of Triple Systems \(I\): Adding One Point

Abstract

We solve the problem of existence of minimal enclosings for triple systems with \(1 \leq \lambda \leq 6\) and any \(v\), i.e., an inclusion of \(\text{BIBD}(v, 3, \lambda)\) into \(\text{BIBD}(v+1, 3, \lambda+m)\) for minimal positive \(m\). A new necessary general condition is derived and some general results are obtained for larger \(\lambda\) values.