# Nonsequenceable Steiner triple systems

## TL;DR

This paper investigates the properties of partial Steiner triple systems, specifically focusing on their sequenceability, and demonstrates the existence of nonsequenceable systems with specific sizes for certain orders.

## Contribution

It introduces the concept of nonsequenceable partial Steiner triple systems and proves their existence for a range of sizes when the order is congruent to 1 mod 6.

## Key findings

- Existence of nonsequenceable PSTS(n) for specified sizes
- Construction methods for nonsequenceable systems
- Results applicable for all n ≡ 1 mod 6, except n=7

## Abstract

A partial Steiner triple system is is $sequenceable$ if the points can be sequenced so that no proper segment can be partitioned into blocks. We show that, if $0 \leq a \leq (n-1)/3$, then there exists a nonsequenceable PSTS$(n)$ of size $\frac{1}{3}\binom{n}{2}-a$, for all $n \equiv 1 \pmod{6}$ except for $n=7$.