Block-avoiding point sequencings of Mendelsohn triple systems

• Published in: arXiv.org
• Main Authors: Kreher, Donald L, Stinson, Douglas R, Veitch, Shannon
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 19.09.2019
• Subjects: Sequences; Upper bounds
• ISSN: 2331-8422

A cyclic ordering of the points in a Mendelsohn triple system of order \(v\) (or MTS\((v)\)) is called a sequencing. A sequencing \(D\) is \(\ell\)-good if there does not exist a triple \((x,y,z)\) in the MTS\((v)\) such that (1) the three points \(x,y,\) and \(z\) occur (cyclically) in that order in \(D\); and (2) \(\{x,y,z\}\) is a subset of \(\ell\) cyclically consecutive points of \(D\). In this paper, we prove some upper bounds on \(\ell\) for MTS\((v)\) having \(\ell\)-good sequencings and we prove that any MTS\((v)\) with \(v \geq 7\) has a \(3\)-good sequencing. We also determine the optimal sequencings of every MTS\((v)\) with \(v \leq 10\).