The Girth of a Design

In 1976 Erdős asked about the existence of Steiner triple systems that lack collections of \(j\) blocks employing just \(j+2\) points. This has led to the study of anti-Pasch, anti-mitre and 5-sparse Steiner triple systems. Simultaneously generating sets and bases for Steiner triple systems and \(t\)-designs have been determined. Combining these ideas, together with the observation that a regular graph is a 1-design, we arrive at a natural definition for the girth of a design. In turn, this provides a natural extension of the search for cages to the universe of all \(t\)-designs. We include the results of computational experiments that give an abundance of examples of these new definitions.