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 $$\mathcal{A}_r={F\in\binom{[n]}k:|F\cap[t+2r]|\ge t+r}$$ be the Frankl families and write $ρ_r$ 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 $ρ_0\geρ_1$; 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 $n\ge2k$.