Block-avoiding point sequencings of Mendelsohn triple systems

TL;DR

This paper investigates special cyclic orderings of points in Mendelsohn triple systems that avoid certain consecutive triples, establishing bounds and exact solutions for small systems.

Contribution

It introduces the concept of $oldsymbol{ extit{ ext{ell}}}$-good sequencings in Mendelsohn triple systems and proves existence results and bounds for these orderings.

Findings

Any MTS(v) with v ≥ 7 has a 3-good sequencing.

Upper bounds on $ extit{ ext{ell}}$ for $ extit{ ext{ell}}$-good sequencings are established.

Optimal sequencings are determined for all MTS(v) with v ≤ 10.

Abstract

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 $v \leq 10$.