Erdős Problem #812 — open problem in combinatorics

Erdős Problem #812

Is it true that R(n+1)/R(n)≥ 1+c for some constant c>0, for all large n?

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.

Start working on it Submit a claim Follow

Cite

@misc{cairn-erdos-812,
  title        = {Erdős Problem #812},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-812}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

erdos_812.parts.i. Is it true that $\frac{R ( n + 1 )}{R ( n )} \geq 1 + c$ for some constant $c > 0$ , for all large $n$ ?

erdos_812.parts.ii. Is it true that $R (n + 1) - R (n) ≫ n^{2}$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«812» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_812.parts.i :
    answer(sorry) ↔ ∃ c > 0, ∀ᶠ n in atTop, (R (n + 1) : ℝ) / (R n : ℝ) ≥ 1 + c

theorem erdos_812.parts.ii :
    answer(sorry) ↔
      (fun n : ℕ ↦ (R (n + 1) : ℝ) - (R n : ℝ)) ≫ (fun n : ℕ ↦ (n : ℝ) ^ 2)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/812. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/812
[BEFS89] Burr, S. A. and Erd\H{o}s, P. and Faudree, R. J. and Schelp, R. H., On the difference between consecutive {R}amsey numbers. Utilitas Math. (1989), 115--118.

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.