[论文解读] Generalized degrees and densities for families of sets

Emanuel Knill|arXiv (Cornell University)|Nov 8, 1994

Mathematical Approximation and Integration被引用 7

一句话总结

本文引入了集合族的广义度与密度概念，针对在结构约束下每个元素至多出现在 k 个集合中的局部 k-宽族，建立了线性上界。证明了中心化且局部 k-宽的族满足 |F| ≤ (k+1)(n−k/2)，且该界为紧致界；并发展了一套针对半格与偏序集的密度框架，推广了并集闭合集猜想，证明了某些格的密度性质。

ABSTRACT

Let F be a family of subsets of {1,2,...,n}. The width-degree of an element x in at least one member of F is the width of the family {U in F | x in U}. If F has maximum width-degree at most k, then F is locally k-wide. Bounds on the size of locally k-wide families of sets are established. If F is locally k-wide and centered (every U in F has an element which does not belong to any member of F incomparable to U), then |F| <= (k+1)(n-k/2); this bound is best possible. Nearly exact bounds, linear in n and k, on the size of locally k-wide families of arcs or segments are determined. If F is any locally k-wide family of sets, then |F| is linearly bounded in n. The proof of this result involves an analysis of the combinatorics of antichains. Let P be a poset and L a semilattice (or an intersection-closed family of sets). The P-size of L is |L^P|. For u in L, the P-density of u is the ratio |[u)^P|/|L^P|. The density of u is given by the [1]-density of u. Let p be the number of filters of P. L has the P-density property iff there is a join-irreducible a in L such that the P-density of a is at most 1/p Which non-trivial semilattices have the P-density property? For P=[1], it has been conjectured that the answer is: "all" (the union-closed sets conjecture). Certain subdirect products of lower-semimodular lattices and, for P=[n], of geometric lattices have the P-density property in a strong sense. This generalizes some previously known results. A fixed lattice has the [n]-density property if n is large enough. The density of a generator U of a union-closed family of sets L containing the empty set is estimated. The estimate depends only on the local properties of L at U. If L is generated by sets of size at most two, then there is a generator U of L with estimated density at most 1/2.

研究动机与目标

• 将集合族中度与密度的概念推广至标准组合设定之外的范畴。
• 在中心化约束下，为局部 k-宽集合族的大小建立紧致的线性上界。
• 通过为半格与偏序集引入 P-密度框架，扩展并集闭合集猜想。
• 分析并集闭合族中生成元的密度，特别是由大小至多为二的集合生成的族。
• 确定哪些半格与格满足特定偏序集 P（包括 [1] 与 [n]）的 P-密度性质。

提出的方法

• 将元素的宽度度定义为包含该元素的集合的最大数量，从而引出局部 k-宽族的概念。
• 为半格 L 中的元素引入 P-大小与 P-密度（P 为偏序集），以推广密度度量。
• 利用反链组合学与滤子结构分析局部 k-宽族的极值大小。
• 证明若一族集合为中心化且局部 k-宽，则其大小被限制在 (k+1)(n−k/2) 以内，且等式可能成立。
• 证明某些下半模格与几何格的子直积在强意义上满足 P-密度性质。
• 通过局部结构特性估计并集闭合族中生成元的密度，表明对于大小 ≤2 的集合，存在某个生成元其密度 ≤1/2。

实验结果

研究问题

• RQ1 在 {1,…,n} 的子集族中，中心化且局部 k-宽族的最大可能大小是多少？
• RQ2 哪些非平凡半格对给定偏序集 P 满足 P-密度性质？
• RQ3 每个有限格是否对充分大的 n 都具有 [n]-密度性质？
• RQ4 能否仅通过生成元处的局部信息来界定并集闭合族中生成元的密度？
• RQ5 广义度与密度如何推广并集闭合集猜想及相关极值集合论问题？

主要发现

• 中心化且局部 k-宽族 F 的大小至多为 (k+1)(n−k/2)，且该界为紧致界。
• 对于弧或线段的局部 k-宽族，建立了接近精确的 n 与 k 的线性上界，证实了线性增长。
• 某些下半模格与几何格的子直积满足 P-密度性质，尤其当 P=[n] 时。
• 每个固定的有限格对所有充分大的 n 都具有 [n]-密度性质。
• 在由大小至多为二的集合生成的并集闭合族中，存在一个生成元其 P-密度至多为 1/2，该结果仅基于局部结构估计得出。
• 猜想 [1]-密度性质对所有并集闭合族成立，从而推广了并集闭合集猜想。