Erdős Problem #1029 — open problem in graph theory

Erdős Problem #1029

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 R(k)/k2^k/2→ ∞.

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-1029,
  title        = {Erdős Problem #1029},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1029}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

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 $\frac{R ( k )}{k 2 ^{k /2}} \to \infty.$

In [Er93] Erdős offers $100 for a proof of this and $1000 for a disproof, but says 'this last offer is to some extent phoney: I am sure that this is true (but I have been wrong before).'

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1029».

theorem erdos_1029 :
    Tendsto (fun k : ℕ ↦ (SimpleGraph.diagonalRamsey k : ℝ) /
      ((k : ℝ) * (2 : ℝ) ^ ((k : ℝ) / 2))) atTop atTop

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/1029. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1029
[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.
[ErSz35] Erdős, P. and Szekeres, G., A combinatorial problem in geometry. Compos. Math. (1935), 463-470.
[Sp75] Spencer, J., Ramsey's theorem - a new lower bound. J. Combin. Theory Ser. A (1975), 108-115.

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.