A subset $I$ of the vertex set $V(G)$ of a graph $G$ is called a $k$-clique independent set of $G$ if no $k$ vertices in $I$ form a $k$-clique of $G$. An independent set is a $2$-clique independent set. Let $\pi_k(G)$ denote the number of $k$-cliques of $G$. For a function $w: V(G) \rightarrow \{0, 1, 2, \dots\}$, let $G(w)$ be the graph obtained from $G$ by replacing each vertex $v$ by a $w(v)$-clique $K^v$ and making each vertex of $K^u$ adjacent to each vertex of $K^v$ for each edge $\{u,v\}$ of $G$. For an integer $m \geq 1$, consider any $w$ with $\sum_{v \in V(G)} w(v) = m$. For $U \subseteq V(G)$, we say that $w$ is uniform on $U$ if $w(v) = 0$ for each $v \in V(G) \setminus U$ and, for each $u \in U$, $w(u) = \left\lfloor m/|U| \right\rfloor$ or $w(u) = \left\lceil m/|U| \right\rceil$. Katona asked if $\pi_k(G(w))$ is smallest when $w$ is uniform on a largest $k$-clique independent set of $G$. He placed particular emphasis on the Sperner graph $B_n$, given by $V(B_n) = \{X \colon X \subseteq \{1, \dots, n\}\}$ and $E(B_n) = \{\{X,Y\} \colon X \subsetneq Y \in V(B_n)\}$. He provided an affirmative answer for $k = 2$ (and any $G$). We determine graphs for which the answer is negative for every $k \geq 3$. These include $B_n$ for $n \geq 2$. Generalizing Sperner's Theorem and a recent result of Qian, Engel and Xu, we show that $\pi_k(B_n(w))$ is smallest when $w$ is uniform on a largest independent set of $B_n$. We also show that the same holds for complete multipartite graphs and chordal graphs. We show that this is not true of every graph, using a deep result of Bohman on triangle-free graphs.

翻译：图$G$的顶点集$V(G)$的子集$I$称为$G$的$k$-团独立集，若$I$中任意$k$个顶点都不构成$G$的一个$k$-团。独立集即为$2$-团独立集。设$\pi_k(G)$表示$G$中$k$-团的个数。对于函数$w: V(G) \rightarrow \{0, 1, 2, \dots\}$，令$G(w)$为将$G$中每个顶点$v$替换为一个$w(v)$-团$K^v$，并对$G$的每条边$\{u,v\}$，使$K^u$中每个顶点与$K^v$中每个顶点相邻后得到的图。对于整数$m \geq 1$，考虑任意满足$\sum_{v \in V(G)} w(v) = m$的$w$。对于$U \subseteq V(G)$，若对每个$v \in V(G) \setminus U$有$w(v) = 0$，且对每个$u \in U$有$w(u) = \left\lfloor m/|U| \right\rfloor$或$w(u) = \left\lceil m/|U| \right\rceil$，则称$w$在$U$上是均匀的。Katona提出如下问题：当$w$在$G$的最大$k$-团独立集上均匀时，$\pi_k(G(w))$是否取最小值？他特别关注Sperner图$B_n$，其中$V(B_n) = \{X \colon X \subseteq \{1, \dots, n\}\}$，$E(B_n) = \{\{X,Y\} \colon X \subsetneq Y \in V(B_n)\}$。他对$k=2$（及任意$G$）给出了肯定回答。我们确定了所有使得对每个$k \geq 3$回答为否定的图，其中包含$n \geq 2$时的$B_n$。通过推广Sperner定理及Qian、Engel和Xu的最新结果，我们证明了当$w$在$B_n$的最大独立集上均匀时$\pi_k(B_n(w))$取最小值。我们还证明了对完全多部图和弦图该结论同样成立。利用Bohman关于无三角形图的深刻结果，我们表明并非所有图都满足该性质。