The forcing conjecture for bipartite graphs
A graph with edges is -forcing if every sequence of graphs with edge density tending to and is quasirandom of density , and forcing if this holds for every . Chung, Graham and Wilson showed the four-cycle is forcing; Skokan and Thoma asked which bipartite graphs can replace it. Conlon, Fox and Sudakov formulated the forcing conjecture: a graph is forcing if and only if it is bipartite and contains a cycle. The conjecture implies Sidorenko's conjecture. Is every bipartite graph containing a cycle forcing?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Quasirandom graphs, graph limits
- Posed by
- Question of J. Skokan and L. Thoma (Graphs Combin., 2004); conjecture formulated by D. Conlon, J. Fox and B. Sudakov (GAFA, 2010, Conjecture 2)
- Year posed
- 2004
- Years open
- 22y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 28 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Corollary 1.3: for the same connected bipartite (which contains cycles) there are and a nonconstant graphon with and ; sampling gives finite graphs with edge and -densities tending to and but four-cycle density tending to a limit above . So is not -forcing at that , and the conjecture fails. Since the forcing conjecture implies Sidorenko's, this follows from the Sidorenko counterexample; no separate idea is claimed.
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The disproof is Corollary 1.3 of the Sidorenko counterexample manuscript, obtained by interpolating between kernels of the same mean and sampling; it is not covered by the Lean formalization.
Verification
No independent mathematician has checked this yet. Checked here: Corollary 1.3 and its proof outline in 'A counterexample to Sidorenko's conjecture', read against the forcing conjecture as quoted from Conlon-Fox-Sudakov. The argument is a short consequence of Theorem 1.1, which is Lean-checked; the corollary itself is not formalized and was not refereed. The manuscript notes it exhibits one density p only and does not claim failure of p-forcing at every density, which is enough to refute the conjecture as stated.