# Sequencing Partial Steiner Triple Systems

## TL;DR

This paper proves that partial Steiner triple systems with up to three point-disjoint blocks can be arranged in a sequence where no segment is a union of blocks, advancing understanding of their sequencing properties.

## Contribution

It establishes a new sufficient condition for the sequenceability of partial Steiner triple systems based on the number of point-disjoint blocks.

## Key findings

- Partial Steiner triple systems with ≤3 point-disjoint blocks are sequenceable.
- Provides a characterization linking block disjointness to sequenceability.
- Advances the theory of sequencing in combinatorial design systems.

## Abstract

A partial Steiner triple system of order n is sequenceable if there is a sequence of length n of its distinct points such that no proper segment of the sequence is a union of point-disjoint blocks. We prove that if a partial Steiner triple system has at most three point-disjoint blocks, then it is sequenceable.