具有有界匹配数的均匀超图的张量谱稳定性
Tensor Spectral Stability for Uniform Hypergraphs with Bounded Matching Number
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中文总结 AI 辅助
研究具有有界匹配数的均匀超图的张量谱稳定性，通过建立定理证明匹配数至多为$\beta$且张量谱半径接近最大值的超图在结构上接近极值超图$S_{n,k,\beta}$，还应用此结果为厄尔多斯匹配猜想谱版本给出新证明。
中文摘要 AI 辅助
我们为具有有界匹配数的均匀超图建立了一个张量谱稳定性定理。具体而言，对于固定整数$k\geq3$和$\beta\geq2$以及足够大的$n$，我们证明每个$n$顶点的$k$均匀超图$H$，其匹配数至多为$\beta$且张量谱半径接近所有此类超图中的最大可能值，在结构上必定接近极值超图$S_{n,k,\beta}$。其边由与固定的$\beta$个顶点集相交的所有$k$集组成。此外，我们表明$H$的每条边都与这个特殊顶点集相交，并且$H$包含$S_{n,k,\beta}$中除一小部分边之外的所有边。作为应用，我们为足够大的$n$获得了厄尔多斯匹配猜想谱版本的新证明。
英文摘要
We establish a tensor spectral stability theorem for uniform hypergraphs with bounded matching number. More precisely, for fixed integers $k\geq 3$ and $β\geq2$, and sufficiently large $n$, we prove that every $n$-vertex $k$-uniform hypergraph $H$ with matching number at most $β$ and tensor spectral radius close to the maximum possible value among all such hypergraphs must be structurally close to the extremal hypergraph $S_{n,k,β}$, whose edges consist of all $k$-sets intersecting a fixed set of $β$ 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 $S_{n,k,β}$. As an application, we obtain a new proof of the spectral version of the Erdős matching conjecture for sufficiently large $n$.
