Prove Sidorenko’s conjecture

Prove that every bipartite graph is 2-common; equivalently, establish Sidorenko’s conjecture that the minimum asymptotic density of monochromatic copies of every bipartite graph in a two-coloured complete graph is at least the density expected in a uniformly random two-colouring.

Background

The paper discusses the notion of a graph being 2-common, meaning that its two-colour Ramsey multiplicity asymptotically matches the random-colouring benchmark. Sidorenko’s conjecture, independently formulated by Sidorenko and by Erdős and Simonovits, would imply that every bipartite graph is 2-common. Although many cases have been proved, 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”