高度为四的Frankl猜想与高度为五的反例结构
Frankl's Conjecture at Height Four and the Structure of Height-Five Counterexamples
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中文总结 AI 辅助
该研究通过并闭族的高度证明了Frankl并闭集猜想在高度≤4时成立，推导了高度≤5的空集反例的结构性质，排除了临界并覆盖数为5的情况，为高度5的反例提供了关键约束。
中文摘要 AI 辅助
我们通过包含偏序集的高度来研究Frankl的并闭集猜想。在等价的不含空集的表述中，即寻找一个元素，它被严格多于一半的成员所包含，我们证明了每个高度至多为四的有限并闭族都满足该猜想。等价地，通常的至少一半表述对于每个包含空集且高度至多为五的并闭族也成立。我们还为下一个未解决的情况建立了结构理论。假设存在一个高度至多为五的最小不含空集的反例，我们证明其基数为偶数2t，至少有三个频率为t的临界元素，且满足最小反例界t≥2n-1。每个临界元素确定一个形如U\{x}的余原子，而每个临界对在完全双回避顶和具有三层迹正规形式的大回避纤维之间满足二分性。坐标删除进一步产生一个精确的匹配缺陷障碍。最后，引入覆盖所有临界元素所需的并不可约成员的最小数量，我们排除了五覆盖的情况，并证明该临界并覆盖数为3或4。这些结果极大地约束了任何可能的高度五反例，同时留下了明确的剩余转移问题。
英文摘要
We study Frankl's union-closed sets conjecture through the height of the inclusion poset. Working in the equivalent empty-set-free formulation, where one seeks an element contained in strictly more than half of the members, we prove the conjecture for every finite union-closed family of height at most four. Equivalently, the usual at-least-half formulation holds for every union-closed family containing the empty set and having height at most five. We also develop a structural theory for the next unresolved case. Assuming a smallest empty-set-free counterexample of height at most five, we show that it has even cardinality $2t$, at least three critical elements of frequency $t$, and satisfies the minimal-counterexample bound $t \geq 2n-1$. Every critical element determines a coatom of the form $U \setminus \{x\}$, while every critical pair satisfies a dichotomy between a full double-avoidance top and a large avoidance fiber admitting a three-layer trace normal form. Coordinate deletion further yields an exact matching-defect obstruction. Finally, introducing the minimum number of join-irreducible members required to cover all critical elements, we exclude the five-cover case and show that this critical join-cover number is either three or four. These results substantially constrain any possible height-five counterexample while leaving the remaining transfer problem explicit.
