Minimum Number of Below Average Triangles in a Weighted Complete Graph

Let G be an edge weighted graph with n nodes, and let A(3,G) be the average weight of a triangle in G. We show that the number of triangles with weight at most equal to A(3,G) is at least (n−2) and that this bound is sharp for all n≥7. Extensions of this result to cliques of cardinality k>3 are also discussed.