Threshold for Steiner triple systems

TL;DR

This paper establishes the threshold probability for the existence of spanning Steiner triple systems and Latin squares in random hypergraphs, using a novel bootstrapping scheme and advanced probabilistic methods.

Contribution

It introduces a new bootstrapping approach combining iterative absorption and fractional expectation-thresholds to determine the threshold for Steiner triple systems.

Findings

Threshold probability for Steiner triple systems established

Threshold for Latin squares also proven

Novel bootstrapping scheme developed

Abstract

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.