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.

中文翻译：

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Steiner 三重系统的阈值

我们以高概率证明\(\mathbb {G}^{(3)}(n,n^{-1+o(1)})\)包含\(n\equiv 1 的跨越斯坦纳三元组， 3\pmod {6}\)，建立斯坦纳三元系统存在的阈值概率的指数。我们还证明了拉丁方的类似定理。我们的结果源自一种新颖的引导方案，该方案利用迭代吸收以及弗兰克斯顿、卡恩、纳拉亚南和帕克建立的阈值和分数期望阈值之间的联系。