The fundamental theorem of Tur\'{a}n from Extremal Graph Theory determines the exact bound on the number of edges $t_r(n)$ in an $n$-vertex graph that does not contain a clique of size $r+1$. We establish an interesting link between Extremal Graph Theory and Algorithms by providing a simple compression algorithm that in linear time reduces the problem of finding a clique of size $\ell$ in an $n$-vertex graph $G$ with $m \ge t_r(n)-k$ edges, where $\ell\leq r+1$, to the problem of finding a maximum clique in a graph on at most $5k$ vertices. This also gives us an algorithm deciding in time $2.49^{k}\cdot(n + m)$ whether $G$ has a clique of size $\ell$. As a byproduct of the new compression algorithm, we give an algorithm that in time $2^{\mathcal{O}(td^2)} \cdot n^2$ decides whether a graph contains an independent set of size at least $n/(d+1) + t$. Here $d$ is the average vertex degree of the graph $G$. The multivariate complexity analysis based on ETH indicates that the asymptotical dependence on several parameters in the running times of our algorithms is tight.

翻译：极值图论中的Turán基本定理确定了不含大小为$r+1$的团的$n$顶点图边数$t_r(n)$的精确上界。我们通过提供一个简单压缩算法，在极值图论与算法之间建立了有趣的联系：该算法可在线性时间内将问题归约——即在给定边数$m \ge t_r(n)-k$的$n$顶点图$G$中寻找大小为$\ell$的团（其中$\ell\leq r+1$）——为在最多$5k$个顶点的图上寻找最大团问题。由此我们得到判定$G$是否包含大小为$\ell$的团的算法，其时间复杂度为$2.49^{k}\cdot(n + m)$。作为新压缩算法的副产物，我们还给出一个时间复杂度为$2^{\mathcal{O}(td^2)} \cdot n^2$的算法，用于判定图中是否包含至少$n/(d+1) + t$个顶点的独立集，其中$d$为图$G$的平均顶点度数。基于指数时间假设（ETH）的多变量复杂度分析表明，我们算法运行时间中对多个参数的渐近依赖关系是紧的。