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Level A · Machine-checkable Hard Graph theory P-erdos-77

Erdős Problem #77

If R(k) is the Ramsey number for K_k, the minimal n such that every 2-colouring of the edges of K_n contains a monochromatic copy of K_k, then find the value of lim_k→ inftyR(k)^1/k. This problem is #3 in Ramsey Theory in the graphs problem collection.

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-erdos-77,
  title        = {Erdős Problem #77},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-77}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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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

If is the Ramsey number for , the minimal such that every -colouring of the edges of contains a monochromatic copy of , then find the value of

This problem is #3 in Ramsey Theory in the graphs problem collection.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«77». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_77 :
    Filter.Tendsto (fun k : ℕ ↦ (SimpleGraph.diagonalRamsey k : ℝ) ^ (1 / (k : ℝ)))
      Filter.atTop (𝓝 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 — check erdosproblems.com/77. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/77
  • [BBCGHMST24] Balister, P. and Bollobás, B. and Campos, M. and Griffiths, S. and Hurley, E. and Morris, R. and Sahasrabudhe, J. and Tiba, M., Upper bounds for multicolour Ramsey numbers. arXiv:2410.17197 (2024).
  • [CGMS23] Campos, Marcelo and Griffiths, Simon and Morris, Robert and Sahasrabudhe, Julian, An exponential improvement for diagonal Ramsey. arXiv:2303.09521 (2023).
  • [Er88] Erdős, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.
  • [Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.
  • [GNNW24] Gupta, P. and Ndiaye, N. and Norin, S. and Wei, L., Optimizing the CGMS upper bound on Ramsey numbers. arXiv:2407.19026 (2024).

Source and licence

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