  We prove that with high probability $\mathbb{G}^{(3)}(n,n^{-1+o(1)})$
contains a spanning Steiner triple system for $n\equiv 1,3\pmod{6}$,
establishing the exponent for the threshold probability for existence of a
Steiner triple system. We also prove the analogous theorem for Latin squares.
Our result follows from a novel bootstrapping scheme that utilizes iterative
absorption as well as the connection between thresholds and fractional
expectation-thresholds established by Frankston, Kahn, Narayanan, and Park.
