Borel completeness of the class of countable Steiner triple systems

Abstract: We show that the isomorphism relation for countable Steiner triple systems is Borel complete, that is, the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Steiner triple systems. To prove it, we construct a faithful Borel reduction from countable graphs to countable Steiner triple systems, that is, a Borel assignment $θ$ that associates every countable graph $G$ with a countable Steiner triple system $θ(G)$ so that $G\cong G'$ if and only if $θ(G)\congθ(G')$. Moreover, $θ$ preserves automorphisms which means that $\mathrm{Aut}(G)\cong\mathrm{Aut}(θ(G))$.