On the Minimal Density of Triangles in Graphs

Abstract

For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving thatwhere$t\df \lfloor 1/(1-\rho)\rfloor$is the integer such that$\rho\in\bigl[ 1-\frac 1t,1-\frac 1{t+1}\bigr]$.