Tensor Spectral Stability for Uniform Hypergraphs with Bounded Matching Number

Abstract

We establish a tensor spectral stability theorem for uniform hypergraphs with bounded matching number. More precisely, for fixed integers $k\ge 3$ and ${\hat{I}}^2\ge 2$, and sufficiently large $n$, we prove that every $n$-vertex $k$-uniform hypergraph $H$ with matching number at most ${\hat{I}}^2$ and tensor spectral radius close to the maximum possible value among all such hypergraphs must be structurally close to the extremal hypergraph $S_{n,k,{\hat{I}}^2}$, whose edges consist of all $k$-sets intersecting a fixed set of ${\hat{I}}^2$ vertices. Furthermore, we show that every edge of $H$ intersects this distinguished vertex set and that $H$ contains all but a small proportion of the edges of $S_{n,k,{\hat{I}}^2}$. As an application, we obtain a new proof of the spectral version of the Erdős matching conjecture for sufficiently large $n$.