{"id":"openai-math/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers","pdf":"https://github.com/openai/math/blob/main/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026","tags":["combinatorics","ramsey-theory","van-der-waerden","extremal-combinatorics"],"score":0,"title":"Quantitative Superexponential Bounds for van der Waerden Numbers","domain":"Combinatorics","family":"160","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.CO","summary":"We prove that there are absolute constants c > 0 and K 0 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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-23","reactions":{},"formalized":true,"family_title":"Superexponential van der Waerden numbers"}