Nordhaus-Gaddum inequalities for the number of cliques in a graph

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(Ḡ)=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 Kₙ yielding graphs G₁,G₂,…,Gᵣ and give bounds on ∑ k(Gᵢ) and ∏ k(Gᵢ).