AI 中文总结
该研究通过Kikuchi矩阵的精确谱界，证明了Feige 2008年提出的超图Moore界猜想，给出了k-均匀超图存在小规模偶覆盖的边数阈值条件，相关谱方法还可应用于随机约束满足问题反驳。
AI 中文摘要
一个k-均匀超图的非空子族若满足每个顶点都包含在偶数条超边中，则称为“偶覆盖”；当k=2时，偶覆盖是边不交的环的并，因此偶覆盖的最小规模是围长在超图中的自然对应。我们证明了Feige在2008年提出的关于超图Moore界的猜想：存在绝对常数A和C（与k无关），使得对任意k≥3以及任意1≤ℓ≤n，任何有n个顶点、超边数超过C n^{k/2}/ℓ^{k/2-1}的k-均匀超图，都包含一个规模至多为A ℓ log(en/ℓ)的偶覆盖。我们的证明基于Kikuchi矩阵的精确谱界，我们认为该谱界本身也具有独立研究价值；我们在一篇配套论文中将其应用于随机约束满足问题的反驳。
英文摘要
A nonempty subfamily of a $k$-uniform hypergraph is an \emph{even cover} if every vertex lies in an even number of its hyperedges; for $k=2$ these are edge-disjoint unions of cycles, so the minimum size of an even cover is the natural hypergraph analogue of girth. We prove Feige's 2008 conjecture on the hypergraph Moore bound: there are absolute constants $A$ and $C$ (independent of $k$) such that for every $k\ge3$ and every $1\le\ell\le n$, any $k$-uniform hypergraph on $n$ vertices with more than $C\,n^{k/2}/\ell^{k/2-1}$ hyperedges contains an even cover of size at most $A\,\ell\log(en/\ell)$. Our proof is based on sharp spectral bounds for Kikuchi matrices, which we expect to be of independent interest; we apply them to the refutation of random constraint satisfaction problems in a companion paper.
Comments14 pages
