A Sharp Spectral Erdős--Ko--Rado Theorem for Uniform Hypergraphs

Abstract

The spectral Erdős--Ko--Rado problem asks for the largest adjacency-tensor spectral radius of a $t$-intersecting $k$-uniform family. Keevash, Lenz and Mubayi proved that, for fixed $k,t$ and sufficiently large $n$, the unique extremal family is a full $t$-star, and asked whether such a theorem extends to all $n$. Let ${\mathcal{A}}_r=\{F\in (\genfrac{}{}{0ex}{}{[n]}{k}):|F\cap [t+2r]|\ge t+r\}$ be the Frankl families and write ${\rho }_r$ for their spectral radii. For $2\le t<k$ and $n>2k-t$, we prove that ${\mathcal{A}}_0$ is spectrally extremal if and only if ${\rho }_0\ge {\rho }_1$; it is unique up to permutation when the inequality is strict, whereas ${\mathcal{A}}_0$ and ${\mathcal{A}}_1$ are both extremal at equality. The layerwise pull used in the Ahlswede--Khachatrian cardinality proof is not applicable here: applied directly, it may decrease the spectral radius. Our proof instead pulls all boundary layers simultaneously and applies Perron tail symmetrization. It follows that ${\mathcal{A}}_0$ is uniquely extremal for $n\ge (t+1)(k-t+1)+\lceil (t+1)\log (t+1)\rceil$; the leading coefficient $t+1$ is best possible for fixed $t$. We also determine all extremal structures for $t=1$ throughout the range $n\ge 2k$.