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arXiv · 2607.26028

A Spectral Proof of the Hypergraph Moore Bound

Alexander Schmidhuber·Matthew B. Hastings
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Abstract

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.

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Alexander Schmidhuber, Matthew B. Hastings. 2026-07-28. A Spectral Proof of the Hypergraph Moore Bound. https://arxiv.org/abs/2607.26028

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