Resolvability of almost every Steiner triple system

Prove that for every order n congruent to 3 modulo 6, almost every Steiner triple system of order n is resolvable, equivalently, that a uniformly random Steiner triple system of such an order has a decomposition into perfect matchings with high probability.

Background

The paper establishes an analogous high-probability resolvability result for random Latin squares, whose associated 3-partite 3-uniform hypergraphs are decomposable into perfect matchings. It then compares this result with the corresponding problem for random Steiner triple systems, which are non-partite designs.

Ferber and Kwan had previously shown that almost every Steiner triple system has many disjoint perfect matchings covering all but a negligible proportion of its triples. The cited conjecture asks whether this approximate resolvability can be strengthened to full resolvability. The authors explicitly state that the conjecture remains beyond the methods developed for random Latin squares and that new ideas appear necessary.

References

Ferber and Kwan conjectured that, if $n\equiv\, 3\mod 6$, then almost every Steiner triple system of order $n$ is resolvable. That is, that the equivalent result to Theorem~\ref{thm:mainLSversion} should hold for Steiner triple systems. It would seem that new ideas are needed, however, to show this.

— Almost every Latin square has a decomposition into transversals  (2501.05438 - Bowtell et al., 9 Jan 2025) in Section 1, Introduction, subsection “Resolvable designs”