Erdős Problem #544 — open problem in graph theory

Erdős Problem #544

Show that R(3,k+1)-R(3,k)→∞ as k→ ∞. A problem of Erdős and Sós. This problem is #8 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.

Start working on it Submit a claim Follow

Cite

@misc{cairn-erdos-544,
  title        = {Erdős Problem #544},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-544}},
  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_544.parts.i. Show that $R (3, k + 1) - R (3, k) \to \infty$ as $k \to \infty$ .

A problem of Erdős and Sós. This problem is #8 in Ramsey Theory in the graphs problem collection.

erdos_544.parts.ii. Similarly, prove or disprove that $R (3, k + 1) - R (3, k) = o (k) .$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«544» (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_544.parts.i :
    Tendsto (fun k : ℕ ↦ (SimpleGraph.classicalRamsey 3 (k + 1) : ℝ) -
      (SimpleGraph.classicalRamsey 3 k : ℝ)) atTop atTop

theorem erdos_544.parts.ii : answer(sorry) ↔
    (fun k : ℕ ↦ (SimpleGraph.classicalRamsey 3 (k + 1) : ℝ) -
      (SimpleGraph.classicalRamsey 3 k : ℝ)) =o[atTop] (fun k : ℕ ↦ (k : ℝ))

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/544. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/544

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.