Sidorenko’s conjecture

Prove Sidorenko’s conjecture that every bipartite graph is 2-common, meaning that its monochromatic density in every red-blue colouring is asymptotically at least the random-colouring expectation.

Background

The paper discusses the notion of a k-common graph, whose monochromatic copy density matches the expected density in a uniformly random k-edge-colouring. Sidorenko’s conjecture would imply that every bipartite graph is 2-common; although many cases are known, the general conjecture remains unresolved.

References

Sidorenko and Erd\H{o}s and Simonovits independently made what is usually known as `Sidorenko's conjecture', which implies that every bipartite $F$ is $2$-common; despite attracting a lot of attention and being proved in several cases, it remains open.

— The semi-inducibility problem  (2501.09842 - Basit et al., 16 Jan 2025) in Section 2.1, “Goodman’s bound and Ramsey theory”