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Level A · Machine-checkable Hard Graph theory P-erdos-165

Erdős Problem #165

Give an asymptotic formula for R(3,k).

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). Erdős offered a prize of $250.

Current state (tracked automatically)

  • erdosproblems.com lists Erdős problem #165 as open (status updated 31 August 2025).
  • Formal Conjectures marks the Lean statement erdos_165 as open.
All tracked facts and sources
FactValueSource
Status of Erdős problem #165 on erdosproblems.comopen
2025-08-31
erdosproblems.com
Formal Conjectures: status of `erdos_165`research openFormal Conjectures
Formal Conjectures: research statements in the file2 open, 2 solvedFormal 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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

Try it with your chatbot Submit a claim Follow (RSS)
Cite 
@misc{cairn-erdos-165,
  title        = {Erdős Problem #165},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-165}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

Also: CITATION.cff · Atom feed of results

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On the literature board
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Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
  3. Paste its answer here — check it and publish it under your name

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.

or , ,

ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Give an asymptotic formula for .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«165».

theorem erdos_165 :
    (fun k : ℕ ↦ (R(3, k) : ℝ)) ~[atTop] erdos_165.answer

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/165. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_165.variants.conjecture — Campos, Jenssen, Michelen and Sahasrabudhe [CJMS25] and Hefty, Horn, King and Pfender [HHKP25] conjecture that R(3,k) ∼ k^2/2log k.

References

  • erdosproblems.com/165
  • [Er61] Erdős, P., Graph theory and probability. II. Canad. J. Math. (1961), 346-352.
  • [Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
  • [Er90b] Erdős, Paul, *Problems and results on graphs and hypergraphs: similarities and differences*. Mathematics of Ramsey theory (1990), 12-28.
  • [Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.
  • [Er97c] Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67.
  • [Sh83] Shearer, J. B., A note on the independence number of triangle-free graphs. Discrete Math. (1983), 83-87.
  • [Ki95] Kim, J. H., The Ramsey number has order of magnitude . Random Structures Algorithms (1995), 173-207.
  • [CJMS25] Campos, M. and Jenssen, M. and Michelen, M. and Sahasrabudhe, J., *A new lower bound for the Ramsey numbers *. arXiv:2505.13371 (2025).
  • [HHKP25] Hefty, Z. and Horn, P. and King, D. and Pfender, F., *Improving in just two bites*. arXiv:2510.19718 (2025).

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.