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Level A · Machine-checkable Hard Combinatorics P-ramsey-numbers

Ramsey numbers

The open problem: determine the Ramsey number R(5,5). It is known that 43 ≤ R(5,5) ≤ 46.

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-ramsey-numbers,
  title        = {Ramsey numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/ramsey-numbers}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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The problem

The question

The open problem: determine the Ramsey number .

It is known that .

The (graph) Ramsey number is the least natural number such that every simple graph on vertices contains either a clique of size or an independent set of size (equivalently, the complement graph contains a clique of size ).

We formalize the classical open problem of determining , together with the currently best known bounds .

Note: the diagonal Ramsey number can also be formulated in terms of 2-colorings of -subsets, as Combinatorics.hypergraphRamsey 2 n (see FormalConjecturesForMathlib/Combinatorics/Ramsey.lean).

Formal statement (Lean 4)

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

theorem ramsey_number_five_five :
    R(5, 5) = answer(sorry)

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

  • Wikipedia: Ramsey number
  • [Rad] S. P. Radziszowski, Small Ramsey Numbers, Electronic Journal of Combinatorics, Dynamic Survey DS1. (Updated periodically.) https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1
  • [Exoo89] G. Exoo, A lower bound for , Journal of Graph Theory 13 (1989), 97–98. DOI: 10.1002/jgt.3190130113
  • [AM24] V. Angeltveit and B. McKay, , arXiv:2409.15709 (2024).
  • OEIS A212954
  • MathWorld: Ramsey Number

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.