左线性树的非消去交集猜想不成立
The Non-Cancelling-Intersections Conjecture Fails for Left-Linear Trees
中文总结 AI 辅助
Amarilli等人提出的非消去交集（NCI）猜想的两种强化中，仅用左线性树时该猜想不成立，研究通过非构造性论证证明存在规模极大的反例。
中文摘要 AI 辅助
非消去交集（NCI）猜想由Amarilli、Monet和Suciu于2024年提出（arXiv:2401.16210），是组合学中的一个开放问题，指出任意集合的并集可通过不相交并集和子集补集，从其代数非消去交集中构造性构建。同篇论文中提出两种正交的可能强化：仅使用左线性树，以及根据莫比乌斯值的符号，非平凡交集仅正使用或仅负使用。本文证明仅使用左线性树时，该猜想不成立（与另一种强化无关），论证为非构造性，证明存在反例，尽管其规模极大。
英文摘要
First formulated by Amarilli, Monet, and Suciu (arXiv:2401.16210, 2024), the Non-Cancelling Intersections (NCI) conjecture is an open problem in combinatorics stating that any set union can be constructively built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In the same paper, two orthogonal possible strengthenings are proposed: using only left-linear trees, and using non-trivial intersections only positively or only negatively depending on the sign of their Möbius value. Here we show that using only left-linear trees, the conjecture is false (independent of the other strengthening). Our argument is non-constructive. We prove the existence of a counterexample, though it is of immense size.
