The even-uniform hypergraph Moore bound
This resolves a long-standing conjecture in extremal combinatorics for even-uniform hypergraphs, providing a tight bound that generalizes the graph Moore bound.
The authors prove Feige's hypergraph Moore bound conjecture for all even k≥4, showing that any k-uniform hypergraph with average degree d has an even cover of size O(n/d^{1/(k-1)}), matching the conjectured bound without polylogarithmic factors.
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 even $k\ge 4$, with no superfluous polylogarithmic factors. 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 a polynomial interpolation method.