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 (AES), leading to a classification of most previously studied hypergraph families with this property. For every AES $r$-graph $F$, we present 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 $\varepsilon_{K_{\ell+1}} = (3\ell^2-\ell)^{-1}$. Based on a result by Chen-Huang-Kanj-Xia, we 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 ETH fails. Furthermore, we establish an algorithm to decide the $K_{\ell+1}$-freeness of an $n$-vertex graph with $\mathrm{ex}(n,K_{\ell+1})-k$ edges in time $(\ell+1)n^2$ for $k \le n/30\ell$ and $\ell \le \sqrt{n/6}$, partially improving upon the recently provided running time of $2.49^k n^{O(1)}$ by Fomin--Golovach--Sagunov--Simonov. Moreover, we show that for every fixed $\delta > 0$, this problem cannot be solved in time $n^{o(\ell)}$ if $k$ is of order $n^{1+\delta}$ unless ETH fails. 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：经典的Andrásfai-Erdős-Sós定理指出：对于$\ell\ge 2$，每个不含$K_{\ell+1}$的$n$顶点图，若其最小度大于$\frac{3\ell-4}{3\ell-1}n$，则必然是$\ell$部图。我们为$r$图（$r \ge 2$）建立了存在Andrásfai-Erdős-Sós型性质（AES）的简单判别准则，从而分类了此前研究的大多数具有该性质的超图族。对于每个AES $r$图$F$，我们提出一个简单算法，可在$O(n^r)$时间内判定任意最小度大于$(\pi(F) - \varepsilon_F)\binom{n}{r-1}$的$n$顶点$r$图是否包含$F$作为子图，其中$\varepsilon_F >0$为常数。特别地，对于完全图$K_{\ell+1}$，可取$\varepsilon_{K_{\ell+1}} = (3\ell^2-\ell)^{-1}$。基于Chen-Huang-Kanj-Xia的结果，我们证明：对于任意固定常数$C > 0$，若将$\varepsilon_{K_{\ell+1}}$替换为$(C\ell)^{-1}$，则该问题无法在$n^{o(\ell)}$时间内求解（除非ETH失效）。此外，我们建立了一个算法，可在$(\ell+1)n^2$时间内判定任意边数为$\mathrm{ex}(n,K_{\ell+1})-k$的$n$顶点图是否包含$K_{\ell+1}$，其中$k \le n/30\ell$且$\ell \le \sqrt{n/6}$；这部分改进了Fomin-Golovach-Sagunov-Simonov近期给出的$2.49^k n^{O(1)}$运行时间。更进一步，我们证明：对于任意固定常数$\delta > 0$，若$k$的量级达到$n^{1+\delta}$，则该问题无法在$n^{o(\ell)}$时间内求解（除非ETH失效）。作为中间步骤，我们证明：对于特定类别的$r$图$F$，若输入$r$图具有$n$顶点和大最小度，则（满射）$F$染色问题可在$O(n^r)$时间内求解——这精炼了此前若干结果。