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),...,p3(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 p3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.