The classical Andr\'{a}sfai--Erd\H{o}s--S\'{o}s Theorem states that for $\ell\ge 2$, every $n$-vertex $K_{\ell+1}$-free graph with minimum degree greater than $\frac{3\ell-4}{3\ell-1}n$ must be $\ell$-partite. We establish a simple criterion for $r$-graphs, $r \geq 2$, to exhibit an Andr\'{a}sfai--Erd\H{o}s--S\'{o}s type property, also known as degree-stability. This leads to a classification of most previously studied hypergraph families with this property. An immediate application of this result, combined with a general theorem by Keevash--Lenz--Mubayi, solves the spectral Tur\'{a}n problems for a large class of hypergraphs. For every $r$-graph $F$ with degree-stability, there is a simple algorithm to decide the $F$-freeness of an $n$-vertex $r$-graph with minimum degree greater than $(\pi(F) - \varepsilon_F)\binom{n}{r-1}$ in time $O(n^r)$, where $\varepsilon_F >0$ is a constant. In particular, for the complete graph $K_{\ell+1}$, we can take $\varepsilon_{K_{\ell+1}} = (3\ell^2-\ell)^{-1}$, and this bound is tight up to some multiplicative constant factor unless $\mathbf{W[1]} = \mathbf{FPT}$. Based on a result by Chen--Huang--Kanj--Xia, we further show that for every fixed $C > 0$, this problem cannot be solved in time $n^{o(\ell)}$ if we replace $\varepsilon_{K_{\ell+1}}$ with $(C\ell)^{-1}$ unless $\mathbf{ETH}$ fails. Furthermore, we apply the degree-stability of $K_{\ell+1}$ to decide the $K_{\ell+1}$-freeness of graphs whose size is close to the Tur\'{a}n bound in time $(\ell+1)n^2$, partially improving a recent result by Fomin--Golovach--Sagunov--Simonov. As an intermediate step, we show that for a specific class of $r$-graphs $F$, the (surjective) $F$-coloring problem can be solved in time $O(n^r)$, provided the input $r$-graph has $n$ vertices and a large minimum degree, refining several previous results.

翻译：经典Andrásfai–Erdős–Sós定理指出：对于 $\ell\ge 2$，任意不含 $K_{\ell+1}$ 的 $n$ 顶点图，若其最小度大于 $\frac{3\ell-4}{3\ell-1}n$，则必为 $\ell$ 部图。我们为 $r \geq 2$ 的 $r$-图建立了一个简单的判别准则，用以刻画具有Andrásfai–Erdős–Sós型性质（即度稳定性）的超图族。这一定理对大多数先前研究过的具有该性质的超图族给出了分类。结合Keevash–Lenz–Mubayi的一个一般性定理，该结果的直接应用解决了一大类超图的谱Turán问题。对于每个具有度稳定性的 $r$-图 $F$，存在一个简单算法，可在 $O(n^r)$ 时间内判定最小度大于 $(\pi(F) - \varepsilon_F)\binom{n}{r-1}$ 的 $n$ 顶点 $r$-图是否不含 $F$，其中 $\varepsilon_F >0$ 为常数。特别地，对于完全图 $K_{\ell+1}$，可取 $\varepsilon_{K_{\ell+1}} = (3\ell^2-\ell)^{-1}$，且除非 $\mathbf{W[1]} = \mathbf{FPT}$，该界在乘法常数因子意义下是紧的。基于Chen–Huang–Kanj–Xia的结果，我们进一步证明：对任意固定常数 $C > 0$，若将 $\varepsilon_{K_{\ell+1}}$ 替换为 $(C\ell)^{-1}$，则除非 $\mathbf{ETH}$ 失效，该问题不能在 $n^{o(\ell)}$ 时间内解决。此外，我们应用 $K_{\ell+1}$ 的度稳定性，在 $(\ell+1)n^2$ 时间内判定大小接近Turán界的图是否不含 $K_{\ell+1}$，部分改进了Fomin–Golovach–Sagunov–Simonov的最新结果。作为中间步骤，我们证明：对于特定类别的 $r$-图 $F$，当输入 $r$-图具有 $n$ 顶点和大最小度时，其（满射）$F$-染色问题可在 $O(n^r)$ 时间内求解，从而改进先前多个结果。