OpenAI · 160-01 · 2026-09-23 · Combinatorics
Quantitative Superexponential Bounds for van der Waerden Numbers
preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026
Family 160. Superexponential van der Waerden numbers.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Resolves Erdős's superexponential-growth question for van der Waerden numbers. If is the least interval length forcing a monochromatic -term progression in every -coloring, then for an absolute , all and sufficiently large , uniformly in . In particular, for each fixed .
We prove that there are absolute constants c > 0 and K0 such that for every and r ≥ 2. Consequently for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 160: Superexponential van der Waerden numbers. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.