A famous theorem of Kirkman says that there exists a Steiner triple system of order n if and only if n ≡ 1,3 mod 6. In 1973, Erdős conjectured that one can find so-called ‘sparse’ Steiner triple systems. Roughly speaking, the aim is to have at most j −3 triples on every set of j points, which would be best possible. (Triple systems with this sparseness property are also referred to as having high girth.) We prove this conjecture asymptotically by analysing a natural generalization of the triangle removal process. Our result also solves a problem posed by Lefmann, Phelps and Rödl as well as Ellis and Linial in a strong form, and answers a question of Krivelevich, Kwan, Loh and Sudakov. Moreover, we pose a conjecture which would generalize the Erdős conjecture to Steiner systems with arbitrary parameters and provide some evidence for this.

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关于局部稀疏 Steiner 三重系统的 Erdős 猜想

柯克曼的一个著名定理说，当且仅当 n ≡ 1,3 mod 6 时，存在一个 n 阶 Steiner 三重系统。1973 年，Erdős 推测可以找到所谓的“稀疏”Steiner 三重系统。粗略地说，目标是在每组 j 个点上最多有 j -3 个三元组，这可能是最好的。（具有这种稀疏特性的三重系统也被称为具有高周长。）我们通过分析三角形去除过程的自然概括来渐近地证明这一猜想。我们的结果还以强形式解决了 Lefmann、Phelps 和 Rödl 以及 Ellis 和 Linial 提出的问题，并回答了 Krivelevich、Kwan、Loh 和 Sudakov 提出的问题。此外，我们提出了一个猜想，将 Erdős 猜想推广到具有任意参数的 Steiner 系统，并为此提供一些证据。