# P3092: Existence and value of the diagonal Ramsey exponential limit

- ID: `P3092`
- Reference: `diagonal-ramsey-exponential-limit`
- Page: https://theoremdb.org/statements/P3092
- Record maturity: Reviewed problem with recorded work

## 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.

### Context

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.

### Problem setup

- **Definition (Diagonal Ramsey number).** \(R(k)=R(k,k)\) is the two-colour diagonal graph Ramsey number.
- **Definition (Exponential growth constant).** The proposed constant is the ordinary limit of the \(k\)-th roots of \(R(k)\).
- **Remark.** Taking \(k\)-th roots isolates the exponential base of diagonal Ramsey growth.

### What counts as a solution

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

## 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. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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.
- The release review checked 2 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): Current cited bounds give \(\sqrt2\le\liminf R(k)^{1/k}\le\limsup R(k)^{1/k}\le3.7992\ldots\). [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): 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](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): Even existence of the limit is unknown, as is its value.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `diagonal-ramsey-exponential-limit`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://www.erdosproblems.com/77
   - 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
   - reference_database; reference source; checked 2026-08-01
   - Source use: original_summary
   - Supplies the maintained formulation, current open-status assessment, and recorded partial results.
   - 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.
2. <a id="reference-2"></a>Paul Erdős, “Some unsolved problems,” Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221–254. Diagonal Ramsey growth problem https://mathscinet.ams.org/mathscinet/article?mr=0177846
   - journal_article; primary source; checked 2026-08-01
   - Source 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.