An Update on the Existence of Kirkman Triple Systems with Steiner Triple Systems as Subdesigns
A Kirkman triple system of order $v$, KTS($v$), is a resolvable Steiner triple system on $v$ elements. I will describe recent progress made on an open problem posed by Doug Stinson, the existence of KTS($v$) which contain as a subdesign a Steiner triple system of order $u$, an STS($u$). In particular, we have now completely settled the extremal case $v = 2u + 1$, for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case $v = 2u + 3$ to (at present) three possible exceptions. In addition, we have results for other cases of the form $v = 2u + w$ and also near $v = 3u$. Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions. There remain a number of interesting and difficult computational problems in finding KTS with STS subdesigns. I will discuss some of the challenges in trying to completely settle the existence of these designs. All of this work is joint work with Peter Dukes.
Bio: Esther Lamken is currently an independent researcher and consultant in San Francisco. She has worked at the Center for Communications Research in both Princeton and La Jolla and held academic positions at several universities including Caltech, Princeton, U of Vermont, Georgia Tech, and U of Waterloo. She completed her Ph.D. from the University of Michigan in Ann Arbor with a thesis directed by Scott Vanstone at the U. of Waterloo.