t-相交族的谱希尔顿-米尔纳-弗兰克尔定理
A Spectral Hilton--Milner--Frankl Theorem for $t$-Intersecting Families
AI总结：
本文针对非平凡t-相交k-均匀族，在1≤t≤k-2且n≥100·2ᵏk⁷的范围，建立谱希尔顿-米尔纳-弗兰克尔定理，确定极值族的谱半径上界及渐近比较关系。
AI中文摘要：
Keevash、Lenz和Mubayi证明了谱埃尔德什-科-罗定理，表明对于足够大的n，完全t-星在所有t-相交k-均匀族中唯一最大化邻接张量谱半径。本文针对非平凡t-相交族，在显式范围1≤t≤k-2且n≥100·2ᵏk⁷的条件下，建立了谱希尔顿-米尔纳-弗兰克尔定理。更准确地说，我们证明对于每个非平凡t-相交k-均匀族F，其谱半径满足ρ(F)≤max{ρ(Hₙ,ₖ,ₜ),ρ(Aₙ,ₖ,ₜ)}，其中Hₙ,ₖ,ₜ和Aₙ,ₖ,ₜ是经典希尔顿-米尔纳-弗兰克尔定理中的两个极值族。此外，仅当达到最大值的极值候选（同构意义下）时等号成立。我们进一步渐近比较这两个候选：对于每个固定的t，方程(t+2)^(x-t-1)(t+1)^(t+1)=(x-t+1)^(x-1)的唯一实解x=xₜ，会随着k变化决定Hₙ,ₖ,ₜ和Aₙ,ₖ,ₜ中哪一个具有更大的渐近谱半径。
英文摘要：
Keevash, Lenz, and Mubayi proved a spectral Erdős--Ko--Rado theorem, showing that, for sufficiently large $n$, the complete $t$-star uniquely maximizes the adjacency-tensor spectral radius among all $t$-intersecting $k$-uniform families. In this paper, we establish a spectral Hilton--Milner--Frankl theorem for nontrivial $t$-intersecting families in the explicit range $1\le t\le k-2$ and $n\ge 100\cdot 2^k k^7$. More precisely, we prove that, for every nontrivial $t$-intersecting $k$-uniform family $\mathcal F$, the spectral radius satisfies \[ ρ(\mathcal F)\le \max\{ρ(\mathcal H_{n,k,t}),ρ(\mathcal A_{n,k,t})\}, \] where $\mathcal H_{n,k,t}$ and $\mathcal A_{n,k,t}$ are the two extremal families appearing in the classical Hilton--Milner--Frankl theorem. Moreover, equality holds only for the extremal candidates attaining the maximum, up to isomorphism. We further compare the two candidates asymptotically. For each fixed $t$, the unique real solution $x=x_t$ of \[ (t+2)^{x-t-1}(t+1)^{t+1}=(x-t+1)^{x-1} \] determines, as $k$ varies, which of $\mathcal H_{n,k,t}$ and $\mathcal A_{n,k,t}$ has the larger asymptotic spectral radius.
