Erdős (Erdos) Problem #87: status in 2026

Erdős Problem #87

Let 0 < ε < 1. Is it true that, if k is sufficiently large, then R(G) > (1-ε)^k R(k) for every graph G with chromatic number χ(G)=k? The restriction ε < 1 excludes negative bases in (1-ε)^k. This problem is #12 in Ramsey Theory in the graphs problem collection.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31).

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #87 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #87 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.

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Cite

@misc{cairn-erdos-87,
  title        = {Erdős Problem #87},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-87}},
  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

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.

The problem

The question

erdos_87.parts.i. Let $0 < ϵ < 1$ . Is it true that, if $k$ is sufficiently large, then $R (G) > (1 - ϵ)^{k} R (k)$ for every graph $G$ with chromatic number $χ (G) = k$ ?

The restriction $ϵ < 1$ excludes negative bases in $(1 - ϵ)^{k}$ .

This problem is #12 in Ramsey Theory in the graphs problem collection.

erdos_87.parts.ii. Even stronger, is there some $c > 0$ such that, for all large $k$ , $R (G) > c R (k)$ for every graph $G$ with chromatic number $χ (G) = k$ ?

This problem is #13 in Ramsey Theory in the graphs problem collection.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«87» (2 statements).

theorem erdos_87.parts.i :
    ∀ ε > (0 : ℝ), ε < 1 → ∀ᶠ k : ℕ in atTop,
      ∀ (V : Type) [Fintype V] (G : SimpleGraph V), G.chromaticNumber = (k : ℕ∞) →
        (SimpleGraph.diagonalGraphRamsey G : ℝ) >
          (1 - ε) ^ k * (SimpleGraph.diagonalRamsey k : ℝ)

theorem erdos_87.parts.ii :
    ∃ c > (0 : ℝ), ∀ᶠ k : ℕ in atTop,
      ∀ (V : Type) [Fintype V] (G : SimpleGraph V), G.chromaticNumber = (k : ℕ∞) →
        (SimpleGraph.diagonalGraphRamsey G : ℝ) > c * (SimpleGraph.diagonalRamsey 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/87. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/87
[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.

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.