Solution to a problem of Katona on counting cliques of weighted graphs

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 $\pik(G)$ 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 $\pik(G(w))$ is smallest when $w$ is uniform on a largest $k$-clique independent set of $G$. He placed particular emphasis on the Sperner graph $Bn$, given by $V(Bn) = {X \colon X \subseteq {1, \dots, n}}$ and $E(Bn) = {{X,Y} \colon X \subsetneq Y \in V(Bn)}$. 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 $Bn$ for $n \geq 2$. Generalizing Sperner's Theorem and a recent result of Qian, Engel and Xu, we show that $\pik(Bn(w))$ 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.