We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater generality by Sanders. Using Gowers' local inverse theorem, our bound is quintuple exponential in the length of the progression. We refine this bound in the colour aspect for three‐term progressions, and combine our arguments with an insight of Lefmann to obtain analogous bounds for the Ramsey numbers of certain non‐linear quadratic equations.

中文翻译：

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关于Brauer配置的Ramsey编号

我们在Brauer的van der Waerden定理的推广中获得了一个双指数界，该定理涉及具有相同颜色和相同差异的级数。这样的结果是桑德斯独立获得的，而且具有更大的普遍性。使用高尔斯的局部逆定理，我们的界限在级数的长度上是五倍指数。我们在颜色方面针对三个条件进行了改进，并将我们的论据与Lefmann的见解相结合，从而获得了某些非线性二次方程的Ramsey数的相似边界。