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Level B · Reproducible Graph theory P-sidorenko-s-conjecture

Sidorenko's Conjecture

A graphon is a symmetric measurable function W : [0,1]^2 → [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) = ∫_[0,1]^V(H) Π_v,w ∈ E(H) W(x_v,x_w) Π_v ∈ V(H) dx_v.

From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-sidorenko-s-conjecture,
  title        = {Sidorenko's Conjecture},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/sidorenko-s-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

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The problem

The problem

A graphon is a symmetric measurable function . Given a graphon and a finite graph , the homomorphism density is defined as For a finite bipartite graph , let denote the least constant for which holds for all graphons , where is the complete graph on two vertices. What is ?

What counts as progress

  • A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
  • A proof that a known construction is optimal, or a better bound on the other side.
  • Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.