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

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.

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.

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

Current 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]

1Records

Record	Kind	Assessment
	Result	Supported
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 →
	Result	Supported
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 →
	Trace	Supported
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 →
	Trace	Reported
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 →
No records match these filters.

Notes and companion materialContext, examples, and computations

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 connectTyped relations and evidence flow

ProblemExistence and value of the diagonal Ramsey exponential limit

Claim 1Current status and exact unresolved remainderin this packetSupported
Route 1Dated source and duplicate auditchecksSupported
Claim 2Strongest checked neighboring resultinformsSupported
Route 2Work at the unresolved boundaryaddressesReported

2See also

Cycle Double Cover Conjecturegraph theory
Is there a truly subcubic algorithm for weighted APSP?graph theory
The Total Coloring Conjecturegraph theory

How to cite

TheoremDB contributors, “Existence and value of the diagonal Ramsey exponential limit,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/diagonal-ramsey-exponential-limit

This page as plain text: diagonal-ramsey-exponential-limit.md

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

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

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance