Result 170, Combinatorics

Sharp logarithmic exponents for off-diagonal Ramsey numbers

For every fixed integer s ≥ 5, proves r(s,t)=ts−1/(log⁡t)s−2+o(1)r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)} as t→∞t\to\infty, determining the logarithmic exponent and matching the classical upper bound at that scale. Here r(s,t)r(s,t) is the least number of vertices forcing an s-clique or a t-vertex independent set.

Lean formalization New or sharp bound

The bigger picture

Why it matters

How large can a network be while avoiding both a small group of mutual acquaintances and a large group of mutual strangers? These manuscripts claim a sharper answer to this basic question about unavoidable patterns.

What changes?

The Ramsey number r(s,t) is the smallest vertex count forcing every graph to contain either s pairwise connected vertices or t pairwise unconnected vertices. The unreviewed manuscripts report that, for every fixed integer s at least 5, as t tends to infinity, r(s,t) equals t raised to the power (s minus 1), divided by log t raised to the power (s minus 2 plus a term tending to zero). This matches the classical upper bound's logarithmic exponent.

What does that help mathematicians do?

For s equal to 5, the claim identifies the denominator's logarithmic power as 3. More generally, it would show that the classical upper bound cannot be improved by any fixed additional power of log t in the denominator. Researchers could thus rule out an entire class of stronger bounds, while still investigating finer factors: the result does not determine an exact leading constant.

Are there practical applications?

The immediate value is foundational: it sharpens extremal graph theory's account of how long complete connection and complete separation can both be avoided. It identifies the scale that future bounds and constructions must respect. The supplied claims concern asymptotic thresholds, not a practical network-design procedure, and do not establish finite-size performance or an efficient construction algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The sharp logarithmic exponent of r(5,t)

September 24, 2026 41 pages

We determine the sharp logarithmic exponent of the off-diagonal Ramsey number r(5,t)r(5,t):

r(5,t)=t4(log⁡t)3+o(1)(t⟶∞).\displaystyle r(5,t)=\frac{t^4}{(\log t)^{3+o(1)}} \qquad (t\longrightarrow\infty).

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026,
  author = {{OpenAI}},
  title = {{The sharp logarithmic exponent of $r(5,t)$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026/paper.pdf}{OAI:The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026}},
  year = {2026}
}

Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers

September 24, 2026 53 pages

For every fixed integer s ≥ 6, we determine the sharp logarithmic exponent of the off-diagonal Ramsey number:

r(s,t)=ts−1(log⁡t)s−2+o(1)(t⟶∞).\displaystyle r(s,t)=\frac{t^{s-1}}{(\log t)^{s-2+o(1)}} \qquad (t\longrightarrow\infty).

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026,
  author = {{OpenAI}},
  title = {{Sharp Logarithmic Exponents for Fixed Off-Diagonal Ramsey Numbers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026/paper.pdf}{OAI:Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/170.md.

Sharp logarithmic exponents for off-diagonal Ramsey numbers

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization determines the sharp logarithmic exponent of the off-diagonal Ramsey number r(5,t)r(5,t). It proves r(5,t)=t4/(log⁡t)3+o(1)r(5,t)=t^4/(\log t)^{3+o(1)} as t→∞t\to\infty. For every ε>0\varepsilon>0, the lower bound t4/(log⁡t)3+εt^4/(\log t)^{3+\varepsilon} holds eventually, while the upper bound is Ct4/(log⁡t)3Ct^4/(\log t)^3 for an absolute C>0C>0. The corresponding logarithmic exponent converges to three along all natural values of tt.

The formalization determines the sharp logarithmic exponent of the off-diagonal Ramsey number for every fixed integer s≥6s\ge6. It proves r(s,t)=ts−1/(log⁡t)s−2+o(1)r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)} as t→∞t\to\infty. More explicitly, for each ε>0\varepsilon>0 the lower bound ts−1/(log⁡t)s−2+εt^{s-1}/(\log t)^{s-2+\varepsilon} holds eventually, while the upper bound is Csts−1/(log⁡t)s−2C_s t^{s-1}/(\log t)^{s-2}. The logarithmic exponent limit is taken along all natural values of tt.

Comparator links

Result Comparator statement
Sharp logarithmic exponent of r(5,t)r(5,t) RamseyFive.lean
Sharp logarithmic exponent for fixed off-diagonal Ramsey numbers SharpLogRamsey.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.