[#P3086] An explicit exponential lower bound for diagonal Ramsey numbers

Neutral schematic of a deterministic circuit drawing a red-blue complete graph without a large monochromatic clique. — The objects and operations appearing in An explicit exponential comparison level for diagonal Ramsey numbers.

Problem. Give a deterministic, explicit construction which, for infinitely many integers \(k\), produces a red-blue edge-colouring of \(K_{N_k}\) with no monochromatic \(K_k\), where \(N_k\ge C^k\) for one absolute constant \(C>1\).

1Context

Known frontier: The maintained record gives explicit constructions whose largest clique or independent set is at most \((\log n)^C\), which remains too large to yield \(N_k\ge C_0^k\). Open boundary: Obtain any fixed exponential base greater than one by an explicit construction.

2Problem setup

Definition 1 (Explicit construction). A deterministic finite description or algorithm outputs the colouring from \(k\), with correctness proved without selecting a favourable random outcome.

Definition 2 (Ramsey-avoiding colouring). Neither colour class contains a complete graph on \(k\) vertices.

Remark 1. The probabilistic method gives a stronger existential size, while known explicit constructions remain subexponential in the required sense.

3What counts as a solution

Specify the construction, prove it avoids monochromatic \(K_k\), and prove \(N_k\ge C^k\) for a fixed \(C>1\) on an infinite sequence of \(k\).

1Status

Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: The maintained record gives explicit constructions whose largest clique or independent set is at most \((\log n)^C\), which remains too large to yield \(N_k\ge C_0^k\). Exact unresolved remainder: Obtain any fixed exponential base greater than one by an explicit construction.^[1][2]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: The maintained record gives explicit constructions whose largest clique or independent set is at most \((\log n)^C\), which remains too large to yield \(N_k\ge C_0^k\). Exact unresolved remainder: Obtain any fixed exponential base greater than one by an explicit construction.^[1][2] Relevance to this problem This is the dated publication status for the canonical target An explicit exponential lower bound for diagonal Ramsey numbers. 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: The maintained record gives explicit constructions whose largest clique or independent set is at most \((\log n)^C\), which remains too large to yield \(N_k\ge C_0^k\). The exact unresolved remainder is: Obtain any fixed exponential base greater than one by an explicit construction. A complete resolution must meet the following acceptance conditions: - Specify the construction, prove it avoids monochromatic \(K_k\), and prove \(N_k\ge C^k\) for a fixed \(C>1\) on an infinite sequence of \(k\). Open full record →
	Result	Supported
claim · Claim 2 The maintained record gives explicit constructions whose largest clique or independent set is at most \((\log n)^C\), which remains too large to yield \(N_k\ge C_0^k\).^[1][2] Relevance to this problem Locates the present research frontier immediately below An explicit exponential lower bound for diagonal Ramsey numbers. 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 maintained record gives explicit constructions whose largest clique or independent set is at most \((\log n)^C\), which remains too large to yield \(N_k\ge C_0^k\). This leaves the following boundary unresolved: Obtain any fixed exponential base greater than one by an explicit construction. 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 An explicit exponential lower bound for diagonal Ramsey numbers 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 #78" constructive Ramsey - explicit construction R(k) > C^k - explicit Ramsey graph logarithmic clique independent set 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 Obtain any fixed exponential base greater than one by an explicit construction. Relevance to this problem Turns the remaining uncertainty in An explicit exponential lower bound for diagonal Ramsey numbers 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: Obtain any fixed exponential base greater than one by an explicit construction. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Specify the construction, prove it avoids monochromatic \(K_k\), and prove \(N_k\ge C^k\) for a fixed \(C>1\) on an infinite sequence of \(k\). 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: The maintained record gives explicit constructions whose largest clique or independent set is at most \((\log n)^C\), which remains too large to yield \(N_k\ge C_0^k\). Exact unresolved remainder: Obtain any fixed exponential base greater than one by an explicit construction.

Equivalent-formulation queries: "Erdős Problem #78" constructive Ramsey; explicit construction R(k) > C^k; explicit Ramsey graph logarithmic clique independent set 2025 2026
Strongest checked neighboring result: The maintained record gives explicit constructions whose largest clique or independent set is at most \((\log n)^C\), which remains too large to yield \(N_k\ge C_0^k\).
Exact unresolved remainder: Obtain any fixed exponential base greater than one by an explicit construction.

How the 4 records connectTyped relations and evidence flow

ProblemAn explicit exponential lower bound for diagonal Ramsey numbers

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, “An explicit exponential lower bound for diagonal Ramsey numbers,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/constructive-exponential-ramsey-lower-bound

This page as plain text: constructive-exponential-ramsey-lower-bound.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 #78, Erdős Problems database (living entry), accessed 2026-08-01. Problem #78, OPEN banner, statement, remarks, and bibliography. Problem #78, 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 #78, Erdős Problems database (living entry), accessed 2026-08-01. Problem #78, OPEN banner, statement, remarks, and bibliography.Source used to assess the problem's recorded status.For An explicit exponential lower bound for diagonal Ramsey numbers: This is the dated publication status for the canonical target An explicit exponential lower bound for diagonal Ramsey numbers.Source named by the research packet.
Paul Erdős, “Some unsolved problems,” Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961), 221–254. Probabilistic Ramsey lower bounds and explicit-construction question. ↗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 An explicit exponential lower bound for diagonal Ramsey numbers: 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.

[#P3086] An explicit exponential lower bound for diagonal Ramsey numbers

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