A family of sets is intersecting if any two sets in the family intersect. Given a graph G and an integer r ≥ 1, let I(r)(G) denote the family of independent sets of size r of G. For a vertex v of G, the family of independent sets of size r that contain v is called an r-star. Then G is said to be r-EKR if no intersecting subfamily of I(r)(G) is bigger than the largest r-star. Let n be a positive integer, and let G consist of the disjoint union of n paths each of length 2. We prove that if 1 ≤ r ≤ n/2, then G is r-EKR. This affirms a longstanding conjecture of Holroyd and Talbot for this class of graphs and can be seen as an analogue of a well-known theorem on signed sets, proved using different methods, by Deza and Frankl and by Bollobás and Leader. Our main approach is a novel probabilistic extension of Katona’s elegant cycle method, which might be of independent interest.

中文翻译：

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长度为 2 路径的并集的 Erdős-Ko-Rado 定理

如果该族中的任何两个集合相交，则该族是相交的。给定一个图 G 和一个整数 r ≥ 1，让 I(r)(G) 表示 G 的大小为 r 的独立集族。对于 G 的顶点 v，包含 v 的大小为 r 的独立集族是称为r星。如果 I(r)(G) 的相交亚族没有比最大的 r-star 大，则称 G 是 r-EKR。令 n 为正整数，令 G 由 n 条长度为 2 的路径的不相交并集组成。我们证明如果 1 ≤ r ≤ n/2，则 G 是 r-EKR。这证实了 Holroyd 和 Talbot 对此类图的长期猜想，并且可以看作是著名的有符号集定理的类比，由 Deza 和 Frankl 以及 Bollobás 和 Leader 使用不同方法证明。我们的主要方法是 Katona 的优雅循环方法的新颖概率扩展，