We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class or resolution class). In fact, we prove a general upper bound on the number of perfect matchings in a Steiner triple system and show that almost all Steiner triple systems essentially attain this maximum. We accomplish this via a general theorem comparing a uniformly random Steiner triple system to the outcome of the triangle removal process, which we hope will be useful for other problems. Our methods can also be adapted to other types of designs; for example, we sketch a proof of the theorem that almost all Latin squares have transversals.

中文翻译：

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几乎所有Steiner三重系统都具有完美的匹配

我们证明，对于任何可被3整除的n，几乎所有n阶Steiner三元系统都具有完美的匹配（也称为并行类或分辨率类）。实际上，我们证明了Steiner三重系统中完美匹配的数量的一般上限，并且表明几乎所有Steiner三重系统都达到了这个最大值。我们通过将一个统一的随机Steiner三重系统与三角形去除过程的结果进行比较的一般定理来完成此任务，我们希望该定理对于其他问题也将是有用的。我们的方法也可以适用于其他类型的设计。例如，我们绘制了一个定理的证明，即几乎所有拉丁方都具有横截面。