  Nordhaus and Gaddum proved sharp upper and lower bounds on the sum and
product of the chromatic number of a graph and its complement. Over the years,
similar inequalities have been shown for a plenitude of different graph
invariants. In this paper, we consider such inequalities for the number of
cliques (complete subgraphs) in a graph $G$, denoted $k(G)$. We note that some
such inequalities have been well-studied, e.g., lower bounds on
$k(G)+k(\overline{G})=k(G)+i(G)$, where $i(G)$ is the number of independent
subsets of $G$, has been come to be known as the study of Ramsey multiplicity.
We give a history of such problems. One could consider fixed sized versions of
these problems as well. We also investigate multicolor versions of these
problems, meaning we $r$-color the edges of $K_n$ yielding graphs
$G_1,G_2,\ldots,G_r$ and give bounds on $\sum k(G_i)$ and $\prod k(G_i)$.
