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On the 3-local profiles of graphs. (English) Zbl 1305.05120
Authors’ abstract: For a graph $$G$$, let $$p_i(G)$$, $$i=0,\ldots,3$$ be the probability that three distinct random vertices span exactly $$i$$ edges. We call $$(p_0(G), \ldots,p_3(G))$$ the 3-local profile of $$G$$. We investigate the set $$\mathcal{S}_3 \subset \mathbb{R}^4$$ of all vectors $$(p_0,\ldots,p_3)$$ that are arbitrarily close to the 3-local profiles of arbitrarily large graphs. We give a full description of the projection of $$\mathcal{S}_3$$ to the $$(p_0,p_3)$$ 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 $$p_0+p_3 \geq \frac{1}{4}$$. We also give a full description of the triangle-free case, i.e. the intersection of $$\mathcal{S}_3$$ with the hyperplane $$p_3=0$$. This planar domain is characterized by an SDP constraint that is derived from Razborov’s flag algebra theory.

MSC:
 05C42 Density (toughness, etc.) 05C75 Structural characterization of families of graphs 05C80 Random graphs (graph-theoretic aspects)
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