An Upper bound on the number of Steiner triple systems

TL;DR

This paper establishes an asymptotic upper bound on the number of Steiner triple systems and related factorizations using the entropy method, conjecturing the bounds are tight.

Contribution

It provides the first asymptotic upper bounds on the count of Steiner triple systems and 1-factorizations, employing the entropy method.

Findings

Upper bound on STS(n) as ((1 + o(1)) (n/e^2))^(n^2/6)

Upper bound on F(n) as ((1 + o(1)) (n/e^2))^(n^2/2)

Conjecture that these bounds are sharp

Abstract

Let STS(n) denote the number of Steiner triple systems on n vertices, and let F(n) denote the number of 1-factorizations of the complete graph on n vertices. We prove the following upper bound. STS(n) <= ((1 + o(1)) (n/e^2))^(n^2/6) F(n) <= ((1 + o(1)) (n/e^2))^(n^2/2) We conjecture that the bound is sharp. Our main tool is the entropy method.