Result 161, Combinatorics

Counterexamples to Sidorenko’s conjecture and the forcing conjecture

Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Random graphs provide a natural benchmark for how often a small pattern should appear inside a larger network. This manuscript claims that a particular bipartite pattern breaks two conjectured connections between pattern counts and randomness.

What changes?

The unreviewed manuscript reports a connected bipartite graph, whose vertices split into two groups with edges only between groups, with 35 vertices and 66 edges. Its homomorphism density, the fraction of vertex maps into a host graph that preserve edges, falls below the host's edge density raised to the 66th power in some finite simple graph. The same pattern also violates the forcing conjecture: at one fixed density, asymptotically matching random edge and pattern densities need not imply quasirandomness.

What does that help mathematicians do?

The first claim rules out a universal random-graph lower bound for bipartite pattern densities. The second exposes a limitation of using pattern counts to certify randomness: for this connected pattern, agreement with the random benchmark can conceal nonrandom structure. Quasirandomness means increasingly random-like edge distribution across large vertex subsets. Any general replacement for these conjectures must therefore restrict the patterns or add hypotheses that exclude this example.

Are there practical applications?

The immediate value is foundational, sharpening the relationship between local pattern counts and global graph structure. The reported counterexample identifies a specific pattern that cannot serve, together with edge density alone, as a universal test of quasirandomness. The supplied sources describe no practical algorithm or deployment resulting from the construction.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A counterexample to Sidorenko's conjecture

September 23, 2026 51 pages

We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one fixed density, asymptotically matching the edge and pattern densities does not imply quasirandomness.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-counterexample-to-Sidorenkos-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{A counterexample to Sidorenko's conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf}{OAI:A-counterexample-to-Sidorenkos-conjecture-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/161.md.

Counterexamples to Sidorenko’s conjecture and the forcing conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Sidorenko's conjecture predicts that every bipartite graph HH has homomorphism density at least the host graph's edge density raised to ∣E(H)∣|E(H)|. The formalization disproves this for the paper's fixed bipartite graph with 3535 vertices and 6666 edges: it constructs a nonempty finite simple host graph with t(H,G)<t(K2,G)66t(H,G)<t(K_2,G)^{66}. The separate forcing-conjecture consequence in the paper is outside this statement.

Comparator links

Result Comparator statement
Counterexample to Sidorenko's conjecture SidorenkoCounterexample.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.