  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.
