On the 3-local profiles of graphs

For a graph G, let pi(G),i=0,.,3 be the probability that three distinct random vertices span exactly i edges. We call (p₀(G),.,p ₃(G)) the 3-local profile of G. We investigate the set S3'R4 of all vectors (p₀,.,p₃) that are arbitrarily close to the 3-local profiles of arbitrarily large graphs. We give a full description of the projection of S3 to the (p₀,p₃) plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality p₀+p₃≥14. We also give a full description of the triangle-free case, i.e. the intersection of S3 with the hyperplane p ₃=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.