Steiner triple systems without parallel classes

TL;DR

This paper constructs an infinite family of Steiner triple systems lacking parallel classes for orders congruent to 3 modulo 6, expanding known examples beyond the previously only two known cases of orders 15 and 21.

Contribution

It introduces a method to generate Steiner triple systems without parallel classes for infinitely many orders congruent to 3 mod 6, which was previously unknown.

Findings

Constructed Steiner triple systems without parallel classes for infinitely many orders.

Extended the known set of such systems beyond orders 15 and 21.

Demonstrated existence for all orders congruent to 3 mod 6, with specific constructions.

Abstract

We construct Steiner triple systems without parallel classes for an infinite number of orders congruent to $3 \pmod{6}$. The only previously known examples have order $15$ or $21$.