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.

References

Graphs for which $C_{\ref{sidorenko}(H) = |E(H)|$ are said to have the Sidorenko property, and the Sidorenko conjecture asserts that all bipartite graphs have this property.

— Mathematical exploration and discovery at scale  (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “Sidorenko’s conjecture” (Section 4.12)

For bipartite $H$, a famous conjecture of Sidorenko, which is known for some but not all bipartite $H$, predicts $g_H(\rho)=\rho{e(H)}$.

— A generalised transference principle  (2608.17982 - Allen et al., 18 Aug 2026) in Section 1, subsection “Applications”, subsection “Turán-type results”

The smallest unresolved instance is the ten-vertex, fifteen-edge graph $H=K_{5,5}\setminus C_{10}$, also called the bipartite M\"obius ladder.

— Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment  (2608.23691 - Chung et al., 24 Aug 2026) in Section 3, subsection “Sidorenko's conjecture”