Erdős (Erdos) Problem #1186: status in 2026

Erdős Problem #1186

Let δ_k be such that in any 2-colouring of 1,…,n there exist at least (δ_k+o(1))n^2 many monochromatic k-term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for δ_k. We read "reasonable bounds" as determining the order of magnitude of δ_k as k → ∞.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2026-04-04).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #1186 as open (status updated 4 April 2026).
Formal Conjectures marks the Lean statement erdos_1186 as open.

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #1186 on erdosproblems.com	open 2026-04-04	erdosproblems.com
Formal Conjectures: status of `erdos_1186`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	2 open, 3 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. 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-1186,
  title        = {Erdős Problem #1186},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1186}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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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.

The problem

The question

Let $δ_{k}$ be such that in any $2$ -colouring of ${1, \dots, n}$ there exist at least $(δ_{k} + o (1)) n^{2}$ many monochromatic $k$ -term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for $δ_{k}$ .

We read "reasonable bounds" as determining the order of magnitude of $δ_{k}$ as $k \to \infty$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1186».

theorem erdos_1186 : δ =Θ[atTop] erdos_1186.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/1186. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_1186.variants.prs_conjecture — Parrilo, Robertson, and Saracino [PRS08] have proved 0.0511≈ 1675/32768≤ δ_3 ≤ 117/2192≈ 0.0533, and conjectured the upper bound is correct.

References

erdosproblems.com/1186
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115. PDF
[PRS08] Parrilo, Pablo A. and Robertson, Aaron and Saracino, Dan, *On the asymptotic minimum number of monochromatic 3-term arithmetic progressions*. J. Combin. Theory Ser. A (2008), 185-192. arXiv:math/0609532

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.