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on Sidorenko's Conjecture (AlphaEvolve problem 26)

The statement

A graphon is a symmetric measurable function $W \colon [0,1]^2 \to [0,1]$ . Given a graphon $W$ and a finite graph $H = (V(H),E(H))$ , the homomorphism density $t(H,W)$ is defined as

t(H,W) = \int_{[0,1]^{V(H)}} \prod_{\{v,w\} \in E(H)} W(x_v,x_w) \prod_{v \in V(H)} dx_v.

For a finite bipartite graph $H$ , let $C(H)$ denote the least constant for which

t(H, W) \geq t(K_2, W)^{C(H)}

holds for all graphons $W$ , where $K_2$ is the complete graph on two vertices. What is $C(H)$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.