Mathematics > Combinatorics
[Submitted on 2 Feb 2021 (v1), last revised 16 Jun 2022 (this version, v2)]
Title:New lower bounds for van der Waerden numbers
View PDFAbstract:We show that there is a red-blue colouring of $[N]$ with no blue 3-term arithmetic progression and no red arithmetic progression of length $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number $w(3,k)$ is bounded below by $k^{b(k)}$, where $b(k) = c \big( \frac{\log k}{\log\log k} \big)^{1/3}$. Previously it had been speculated, supported by data, that $w(3,k) = O(k^2)$.
Submission history
From: Ben Green [view email][v1] Tue, 2 Feb 2021 15:16:58 UTC (52 KB)
[v2] Thu, 16 Jun 2022 14:47:38 UTC (53 KB)
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