An Update on the Existence of Kirkman Triple Systems with Subdesigns

TL;DR

This paper advances the understanding of Kirkman triple systems by providing new constructions, settling the extremal case, and introducing a novel method involving Kirkman frames with group divisible design subsystems.

Contribution

It introduces new constructions for KTS with subdesigns, resolves the extremal case v=2u+1, and presents a novel method using Kirkman frames with subsystems.

Findings

Complete resolution of the extremal case v=2u+1.

First infinite classes for the general problem.

Reduced the maximal case v=2u+3 to three exceptions.

Abstract

A Kirkman triple system of order $v$, KTS$(v)$, is a resolvable Steiner triple system on $v$ elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS$(v)$ which contain as a subdesign a Steiner triple system of order $u$, an STS$(u)$. We present several different constructions for designs of this form. As a consequence, we completely settle 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 now three possible exceptions. In addition, we obtain 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.