A simple graph on $n$ vertices may contain a lot of maximum cliques. But how many can it potentially contain? We will show that the maximum number of maximum cliques is taken over so-called cliqueful graphs, more specifically, later we will show that it is taken over saturated composite cliqueful graphs, if $n \ge 15$. Using this we will show that the graph that contains $3^{\lfloor n/3 \rfloor}c$ maxcliques has the most number of maxcliques on $n$ vertices, where $c\in\{1,\frac{4}{3},2\}$, depending on $n \text{ mod } 3$.

翻译：在 $n$ 个顶点上的简单图可能包含大量最大团。但它最多能包含多少个？我们将证明，最大团的最大数量由所谓的团图（cliqueful graphs）取得，更具体而言，当 $n \ge 15$ 时，它由饱和复合团图取得。利用这一结论，我们将证明，包含 $3^{\lfloor n/3 \rfloor}c$ 个最大团的图在 $n$ 个顶点上具有最多的最大团数量，其中 $c\in\{1,\frac{4}{3},2\}$，具体值取决于 $n \text{ mod } 3$。