AI 中文总结
该研究计数函数有向图的后继闭子集，推导生成函数递归公式，确定极值图，定量加强Bollobás集合对不等式，还扩展至其斜变体。
AI 中文摘要
函数有向图是每个顶点出度至多为1的有向图。我们研究函数有向图的后继闭子集的数量，即没有边从该子集流出的子集，证明该分解可得到对应生成函数的简单递归公式。利用该公式，我们在顶点数和边数固定的所有函数有向图中，确定了同时最大化所有大小后继闭子集数量的图。出乎意料的是，该极值结果对Bollobás的集合对不等式给出了定量加强：不仅保证足够大的族中必有一对违反该定理的假设，还证明族的均匀随机子集以高概率见证这种违反，且定量关联于族规模超出经典阈值的程度。我们进一步表明，该方法适用于Hegedus和Frankl提出的Bollobás不等式的斜变体，可得到类似的概率性加强。
英文摘要
A functional digraph is a directed graph where each vertex has an out-degree of at most 1. We study the number of successor-closed subsets of a functional digraph, that is, subsets from which no edge leaves. We show that functional digraphs have a simple recursive formula for their corresponding generating function. Using this formula, we determine, among all functional digraphs with a fixed number of vertices and edges, the one that maximizes and the one that minimizes the number of successor-closed subsets of every size simultaneously. Somewhat unexpectedly, this extremal result yields a quantitative strengthening of the set-pairs inequality of Bollobas: rather than merely guaranteeing that some pair of a large enough family must violate the hypothesis of the theorem, we show that a uniformly random subset of the family witnesses a violation with high probability, quantitatively in terms of how far the family size exceeds the classical threshold. We further show that the same approach applies to the skew variant of Bollobas's inequality due to Hegedus and Frankl, yielding an analogous probabilistic strengthening.
Comments22 pages, 6 figures
