[#P3092] Existence and value of the diagonal Ramsey exponential limit

Problem. Let \(R(k)\) be the least integer \(N\) such that every red-blue colouring of the edges of \(K_N\) contains a monochromatic \(K_k\). Determine whether \(\lim_{k\to\infty}R(k)^{1/k}\) exists and, if it exists, determine its value.

Neutral schematic of a sequence of red-blue complete graphs with a root-scale growth gauge. — The objects and operations appearing in Existence and value of the diagonal Ramsey exponential limit.

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1Context

Known frontier: Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\). Open boundary: Even existence of the limit is unknown, as is its value.

2Problem setup

Definition 1 (Diagonal Ramsey number). \(R(k)=R(k,k)\) is the two-colour diagonal graph Ramsey number.

Definition 2 (Exponential growth constant). The proposed constant is the ordinary limit of the \(k\)-th roots of \(R(k)\).

Remark 1. Taking \(k\)-th roots isolates the exponential base of diagonal Ramsey growth.

3What counts as a solution

Prove convergence of \(R(k)^{1/k}\) and identify the limit, or prove that the sequence has no limit.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\). Exact unresolved remainder: Even existence of the limit is unknown, as is its value.^[1][2]

1Packet records

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\). Exact unresolved remainder: Even existence of the limit is unknown, as is its value.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\). Exact unresolved remainder: Even existence of the limit is unknown, as is its value.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Existence and value of the diagonal Ramsey exponential limit. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\). The exact unresolved remainder is: Even existence of the limit is unknown, as is its value. A complete resolution must meet the following acceptance conditions: - Prove convergence of \(R(k)^{1/k}\) and identify the limit, or prove that the sequence has no limit. Open full record →
Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\).	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\).^[1][2] Relevance to this problem Locates the present research frontier immediately below Existence and value of the diagonal Ramsey exponential limit. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\). This leaves the following boundary unresolved: Even existence of the limit is unknown, as is its value. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there.^[1][2] Relevance to this problem Documents why Existence and value of the diagonal Ramsey exponential limit was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across source collection and its cited primary literature, Crossref/arXiv or MathSciNet stable records, TheoremDB published, prospecting, formal-conjecture, packet, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - "Erdős Problem #77" Ramsey limit - limit R(k)^(1/k) - diagonal Ramsey exponential growth constant 2025 2026 The exact formulation, named variants, 2025–2026 updates, and repository-wide semantic duplicates were checked on 2026-08-01. The source collection still marks the stated remainder open. Living-database status remains subject to later literature not indexed there. Related: Claim 1 Open full record →
Even existence of the limit is unknown, as is its value.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 Even existence of the limit is unknown, as is its value. Relevance to this problem Turns the remaining uncertainty in Existence and value of the diagonal Ramsey exponential limit into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: Even existence of the limit is unknown, as is its value. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Prove convergence of \(R(k)^{1/k}\) and identify the limit, or prove that the sequence has no limit. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\). Exact unresolved remainder: Even existence of the limit is unknown, as is its value.

Equivalent-formulation queries: "Erdős Problem #77" Ramsey limit; limit R(k)^(1/k); diagonal Ramsey exponential growth constant 2025 2026
Strongest checked neighboring result: Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\).
Exact unresolved remainder: Even existence of the limit is unknown, as is its value.

How the 4 records connect

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ProblemExistence and value of the diagonal Ramsey exponential limit

All 3 recorded relations between these records and the problem

2See also

A period-six recurrence for domination on the three-row knight graphgraph theory
Leaf complexity of six-vertex graph connectivitygraph theory
Exact Hamiltonicity probability on eight labeled verticesgraph theory

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Plain text

“Existence and value of the diagonal Ramsey exponential limit.” TheoremDB. P3092. Problem statement; statement text SHA-256 956028bf6948078d1a49568c95e31e52e6661d892eafe8d6a95e90acc61a1f08. https://theoremdb.org/statement/?ref=P3092

BibTeX

@misc{theoremdb-problem-956028bf6948078d1a49568c95e31e52e6661d892eafe8d6a95e90acc61a1f08,
  title = {{Existence and value of the diagonal Ramsey exponential limit}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 956028bf6948078d1a49568c95e31e52e6661d892eafe8d6a95e90acc61a1f08},
  url = {https://theoremdb.org/statement/?ref=P3092}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from erdosproblems.com^[1], current as of August 1, 2026.

1Lean verification

1References

Packet source. Thomas F. Bloom, Erdős Problem #77, Erdős Problems database (living entry), accessed 2026-08-01. Problem #77, OPEN banner, statement, remarks, and bibliography. Problem #77, OPEN banner, statement, remarks, and bibliography. ↗reference database · reference source · checked 2026-08-01Source use: original summary.Supplies the maintained formulation, current open-status assessment, and recorded partial results.Also cited at Thomas F. Bloom, Erdős Problem #77, Erdős Problems database (living entry), accessed 2026-08-01. Problem #77, OPEN banner, statement, remarks, and bibliography.Source used to assess the problem's recorded status.For Existence and value of the diagonal Ramsey exponential limit: This is the dated publication status for the canonical target Existence and value of the diagonal Ramsey exponential limit.Source named by the research packet.
Paul Erdős, “Some unsolved problems,” Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221–254. Diagonal Ramsey growth problem. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Records an original formulation or early published statement of the problem.Source used to assess the problem's recorded status.For Existence and value of the diagonal Ramsey exponential limit: Records an original formulation or early published statement of the problem.

Original TheoremDB statement and summary based on citation-only scholarly sources; no source prose, proof, table, code, or figure is reproduced.

[#P3092] Existence and value of the diagonal Ramsey exponential limit

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