Maximizing the number of Pasch configurations in a Steiner triple system

Grannell, Mike and Lovegrove, Graham (2013). Maximizing the number of Pasch configurations in a Steiner triple system. Bulletin of the Institute of Combinatorics and its Applications, 69 pp. 23–35.

Abstract

Let $P(v)$ denote the maximum number of Pasch configurations in any Steiner triple system on $v$ points. It is known that $P(v)\le M(v)=v(v-1)(v-3)/24$ , with equality if and only if $v$ is of the form $2^n-1$ . It is also known that $\displaystyle \limsup_{v\to\infty \atop v\ne 2^n-1} \frac{P(v)}{M(v)}=1$ . We give a new proof of this result and improved lower bounds on $P(v)$ for certain values of $v$ .