Sidorenko’s conjecture (graph homomorphism densities)

Prove that for every finite bipartite graph H, the inequality t(H,W) ≥ t(K_2,W)^{|E(H)|} holds for all graphons W, i.e., that the optimal constant C_sidorenko(H) equals |E(H)|.

Background

Sidorenko’s conjecture predicts a universal lower bound on homomorphism densities for bipartite graphs in terms of edge count, extending known cases (e.g., complete bipartite graphs, trees, even cycles).

It occupies a central role in extremal combinatorics and graph limits, with deep ties to analytic and probabilistic methods.