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 $w(3,k) = O(k^2)$.

中文翻译：

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范德瓦尔登数的新下界

我们证明存在$[N]$的红蓝色着色，没有蓝色的三项算术级数，也没有长度为 $e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$。因此，双色范德瓦尔登数$w(3,k)$的下界为$k^{b(k)}$，其中$b(k) = c \big ( \frac {\log k }{\log \log k} \big )^{1/3}$。以前有人推测，有数据支持，$w(3,k) = O(k^2)$。