AI 中文总结
研究Feige提出的超图摩尔界猜想，通过在由超图构建的菊池图中利用有色路径，采用多项式插值方法，给出了所有偶数k≥4时该猜想的简单证明，无多余多对数因子。
AI 中文摘要
Feige（2008年）提出的超图摩尔界根据超边的平均密度控制k均匀超图中最小偶数覆盖的大小。偶数覆盖是一组超边，每个顶点被覆盖偶数次，这是图中圈概念的推广，所以最小非平凡偶数覆盖的大小提供了超图围长的概念。从Guruswami、Kothari和Manohar（2022年）的突破性结果开始的近期工作证明了该猜想，误差在多对数因子范围内，其指数后来逐渐得到改进。我们给出了对于所有偶数k≥4的Feige原始超图摩尔界猜想的简单证明，没有多余的多对数因子。我们的证明大致遵循图摩尔界的证明，但使用从超图构建的菊池图中的有色路径，并使用多项式插值方法控制它们的增长。
英文摘要
The hypergraph Moore bound conjectured by Feige (2008) controls the size of the smallest even cover in a $k$-uniform hypergraph in terms of the average density of hyperedges. An even cover is a set of hyperedges covering each vertex an even number of times, generalizing the notion of a cycle in a graph, so the size of the smallest non-trivial even cover provides a notion of hypergraph girth. Recent work, starting from the breakthrough result of Guruswami, Kothari, and Manohar (2022) proved the conjecture up to polylogarithmic factors, whose exponents were later gradually improved. We give a simple proof of Feige's original hypergraph Moore bound conjecture for all $k \geq 3$, with no superfluous polylogarithmic factors. For the case of $k$ even, our proof roughly follows the proof of the graph Moore bound, but works with colored walks in a Kikuchi graph built from a hypergraph and controls their growth using the polynomial method. The argument is then extended to the case of $k$ odd by adapting a procedure in [GKM22].
Commentsv2 also addresses odd-uniform hypergraphs
