Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 160-01. Quantitative Superexponential Bounds for van der Waerden Numbers
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OpenAI@2026-10-07#rem:oai-p-160-01
Authors: OpenAI. Family 160.
Title: Quantitative Superexponential Bounds for van der Waerden Numbers.
Abstract: We prove that there are absolute constants c > 0 and K0 such that $W_r(k){}gt k{}{ck{}lfloor{}log_2 r{}rfloor}$ for every $k{}ge K_0$ and r ≥ 2. Consequently $W_r(k){}{1/k}{}to{}infty$ for each fixed r ≥ 2, giving a quantitative positive resolution of Erdős's superexponential-growth question, including the two-color case.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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