在 $n=2k+1$ 和 $n=2k+2$ 时的典型相交族

中文摘要 AI 辅助

一个集合族被称为相交族，如果其中任意两个成员都相交；如果所有成员都包含一个共同元素，则称为平凡相交族。我们确定了当 $k\to\infty$ 时，在 $2k+1$ 和 $2k+2$ 个元素上的 $k$-均匀相交族的典型结构。对于 $n=2k+2$，我们证明了几乎所有相交族都是平凡的，并且其数量为 \\[ (2k+2+o(1))\\,2^{\binom{2k+1}{k-1}}。\\] 结合 Yang 最近关于 $n\ge 2k+3$ 的结果，这解决了 Balogh、Garcia、Li 和 Wagner 的一个猜想。对于 $n=2k+1$，几乎所有相交族都是非平凡的。我们证明了，正如这几位作者所猜想的那样，一个典型的相交族接近于一个完全星形：其星形之外的成员在连接那些在 $k-1$ 个元素上相交的集合的图中形成大小至多为 2 的分量。我们还得到了这种情况下相交族数量的渐近公式，并在指数中给出了显式的二阶项。我们的证明结合了 Sapozhenko 的图容器方法和 Kneser 图中的稳定性，以控制远离每个星形的族，以及一个聚合物模型和簇展开，以枚举接近固定星形的族。

英文摘要

A family of sets is intersecting if every two members intersect, and trivial if all its members contain a common element. We determine the typical structure of $k$-uniform intersecting families on $2k+1$ and $2k+2$ elements as $k\to\infty$. For $n=2k+2$, we prove that almost all intersecting families are trivial and that their number is \[ (2k+2+o(1))\,2^{\binom{2k+1}{k-1}}. \] Together with Yang's recent result for $n\ge 2k+3$, this settles a conjecture of Balogh, Garcia, Li, and Wagner. For $n=2k+1$, almost all intersecting families are nontrivial. We prove that, as conjectured by the same authors, a typical intersecting family is close to a full star: its members outside the star form components of size at most two in the graph joining sets that intersect in $k-1$ elements. We also obtain an asymptotic formula for the number of intersecting families in this case, with an explicit second-order term in the exponent. Our proof combines Sapozhenko's graph container method and stability in Kneser graphs to control families far from every star, and a polymer model and cluster expansion to enumerate families close to a fixed star.

发表机构

• University of Mississippi（密西西比大学）

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