= Clique counting from iterated shadows
{title2=$\partial^{r-2}\mathcal C\subseteq E(G)$}
For the <uniform set family> of vertex sets of $r$-cliques, its $(r-2)$-fold <lower shadow> consists of graph edges. Repeated use of the <Kruskal-Katona theorem> bounds clique count from edge count. Sixteen four-cliques have at least twenty-three triples in their shadow and at least eighteen edges in their second shadow, by the expansions $16=\binom64+\binom33$ and $23=\binom63+\binom32$. Hence a graph with fifteen edges has at most fifteen four-cliques, attained by the <complete graph> on six vertices.
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