On the Number of Monochromatic 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(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 G₁, G₂, …, G_r and give bounds on ∑ k(G_i) and ∏ k(G_i).