A Spectral Confirmation of the Erd\H{o}s Matching Conjecture

Abstract

The Erd\H{o}s Matching Conjecture concerns the maximum number of hyperedges in an $r$-uniform hypergraph with bounded matching number. In this paper, we study a spectral counterpart of this conjecture. For sufficiently large $n$, we determine the maximum spectral radius over all $n$-vertex $r$-uniform hypergraphs whose matching number is less than $s$, and characterize the unique extremal hypergraph. To establish the main theorem, we first apply the shifting method to reduce the problem to shifted hypergraphs. We then derive several spectral upper bounds through hypergraph decomposition and related variational estimates for tensor spectral radii. With these estimates, we analyze the structural properties of shifted-saturated hypergraphs and prove the spectral extremal theorem for shifted hypergraphs with bounded matching numbers. Finally, we drop the shifted condition and extend our spectral bound to general $r$-uniform hypergraphs. Our main theorem states that for any $n$-vertex $r$-uniform hypergraph $H$ with matching number $\nu(H)<s$, the inequality $\rho(H)\leq \rho(\mathcal{F}_{s-1}(n))$ holds whenever $n$ is sufficiently large. Here $\mathcal{F}_{a}(n)$ denotes the family of all $r$-subsets of $[n]$ intersecting the vertex set $[a]$, and equality is attained if and only if $H$ is isomorphic to $\mathcal{F}_{s-1}(n)$. As an immediate corollary, we derive a spectral counterpart of the classical Erd\H{o}s-Ko-Rado theorem for intersecting hypergraph families.