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Level A · Machine-checkable Hard Combinatorics P-green-38

Green's Open Problem 38

Can we improve the best upper bound? The base c must be positive, since =O compares norms.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-green-38,
  title        = {Green's Open Problem 38},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-38}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Can we improve the best upper bound? The base c must be positive, since =O compares norms.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«38». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_38.upper :
    let ans := (answer(sorry) : ℕ → ℝ)
    LargestAdmissibleCardinality ≤ᶠ[atTop] ans ∧
    ∃ c : ℝ, 0 < c ∧ c < C₂ ∧ ans =O[atTop] (fun n ↦ c ^ n)

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • 100 open problems
  • [La79] Lovász, László. "On the Shannon capacity of a graph." IEEE Transactions on Information theory 25.1 (1979): 1-7.
  • [Po20] Polak, Sven. "New methods in coding theory: Error-correcting codes and the Shannon capacity." arXiv preprint arXiv:2005.02945 (2020).
  • [IRCR26] Itty, Nathaniel and Rosin, Christopher D. and Carstensen, Chase and Reichman, Daniel. "Improved lower bounds for the Shannon capacity of odd cycles." arXiv:2607.21517 (2026).
  • [Ga26] Gao, Yu. "A recursive construction improving the lower bound on the Shannon capacity of ." arXiv:2607.27869 (2026).
  • [BPZ26] Buys, Pjotr and Polak, Sven and Zuiddam, Jeroen. "Lean-verified lower bounds for the Shannon capacity of odd cycles." arXiv:2607.29681 (2026). Lean formalisation: https://github.com/spectra-research/shannon-capacity-lean

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.