Isomorphisms of Infinite Steiner Triple Systems

Abstract

An infinite countable Steiner triple system is called universal if any countable Steiner triple system can be embedded into it. The main result of this paper is the proof of non-existence of a universal Steiner triple system.

The fact is proven by constructing a family $\mathcal{S}$ of size $2^{\omega }$ of infinite countable Steiner triple systems so that no finite Steiner triple system can be embedded into any of the systems from $\mathcal{S}$ and no infinite countable Steiner triple system can be embedded into any two of the systems from $\mathcal{S}$ (it follows that the systems from $\mathcal{S}$ are pairwise non-isomorphic).

A Steiner triple system is called rigid if the only automorphism it admits is the trivial one — the identity. An additional result presented in this paper is a construction of a family of size $2^{\omega }$ of pairwise non-isomorphic infinite countable rigid Steiner triple systems.