Threshold for Steiner triple systems

Abstract

Abstract We prove that with high probability

$$\mathbb {G}^{(3)}(n,n^{-1+o(1)})$$

G

( 3 )

( n ,

n

- 1 + o ( 1 )

)

contains a spanning Steiner triple system for

$$n\equiv 1,3\pmod {6}$$

n ≡ 1 , 3

( mod

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.