 Sidorenko conjecture

The Sidorenko conjecture is the assertion that every finite bipartite graph $H$ satisfiesfor every nonempty finite simple graph $G$. Here $t(H,G)$ is the homomorphism density: the probability that a uniformly random map from $V(H)$ to $V(G)$ sends every edge to an edge. The host graph $G$ need not be bipartite. In the equivalent graphon formulation, the edge constraints are integrated against one symmetric measurable kernel.^[ref]

For the three-vertex graph path, writing $n=|V(G)|$ givesby the Cauchy-Schwarz inequality. The example explains the comparison: correlations between adjacent edge constraints can increase the homomorphism count above the independent-edge heuristic. Bipartiteness is necessary for an assertion covering all hosts, since a nonbipartite $H$ has no homomorphism to a bipartite graph of positive edge density.