Result 160, Combinatorics

Superexponential van der Waerden numbers

Resolves Erdős's superexponential-growth question for van der Waerden numbers. If Wr(k)W_r(k) is the least interval length forcing a monochromatic k-term progression in every r-coloring, then Wr(k)>kck⌊log⁡2r⌋W_r(k)\gt k^{ck\lfloor\log_2 r\rfloor} for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, Wr(k)1/k→∞W_r(k)^{1/k}\to\infty for each fixed r.

Lean formalization Proof

The bigger picture

Why it matters

How long must a row of colored integers be before an evenly spaced pattern of one color becomes unavoidable? The manuscript reports that this threshold grows faster than any fixed-base exponential in the pattern length.

What changes?

The van der Waerden number for r colors and length k is the smallest interval size forcing a same-color, evenly spaced sequence of k integers in every coloring. The unreviewed manuscript claims it exceeds k raised to the power c times k times the integer part of log base two of r. The positive constant c and the minimum allowed k are absolute: the bound holds for every r at least two, with no r-dependent threshold.

What does that help mathematicians do?

For any fixed number of colors, even intervals whose length grows like a fixed constant raised to k can still be colored to avoid a same-color k-term progression, once k is sufficiently large. Thus no fixed-base exponential can serve as a general upper bound. This includes two colors and quantifies how much room there is for colorings that evade these patterns.

Are there practical applications?

Its immediate value is foundational: it sharpens the boundary between unavoidable arithmetic structure and colorings that avoid it. Researchers seeking upper bounds must account for this superexponential obstruction. The stated result is a lower bound, not an efficient coloring algorithm or a demonstrated practical application.

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.

Manuscript

Quantitative Superexponential Bounds for van der Waerden Numbers

September 23, 2026 25 pages Main result formalized in Lean

We prove that there are absolute constants c > 0 and K0 such that Wr(k)>kck⌊log⁡2r⌋W_r(k)\gt k^{ck\lfloor\log_2 r\rfloor} for every k≥K0k\ge K_0 and r ≥ 2. Consequently Wr(k)1/k→∞W_r(k)^{1/k}\to\infty for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026,
  author = {{OpenAI}},
  title = {{Quantitative Superexponential Bounds for van der Waerden Numbers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/paper.pdf}{OAI:Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/160.md.

Superexponential van der Waerden numbers

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

Scope

Let W(r,k)W(r,k) be the least interval length forcing a monochromatic kk-term arithmetic progression in every coloring with at most rr colors. The formalized result gives an absolute threshold KK such that W(r,k)>kk⌊log⁡2r⌋/100000W(r,k)>k^{k\lfloor\log_2 r\rfloor/100000} for all k≥Kk\ge K and r≥2r\ge2. It also establishes the associated growth limits, finiteness, and boundary values. The sharper intermediate estimates used in the paper are not part of the described formalization.

Comparator links

Result Comparator statement
Uniform van der Waerden lower bound QuantitativeVanDerWaerden.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 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.