The girth of a design
Journal of Combinatorial Mathematics and Combinatorial Computing
Mathematics and Computer Science
In 1976 Erdös 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.
Beezer, R A. "The Girth of a Design." Journal of Combinatorial Mathematics and Combinatorial Computing. 40 (2002): 97-114. Print.