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.
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #544 as open (status updated 31 August 2025).
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #544 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: research statements in the file | 2 open, 0 solved | Formal Conjectures |
Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.
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.
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 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/erdos-544)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
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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
erdos_544.parts.i. Show that as .
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
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«544» (2 statements).
theorem erdos_544.parts.i :
Tendsto (fun k : ℕ ↦ (SimpleGraph.classicalRamsey 3 (k + 1) : ℝ) -
(SimpleGraph.classicalRamsey 3 k : ℝ)) atTop atToptheorem erdos_544.parts.ii :
(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
Source and licence
Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.