We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado’s characterisation of a partition regular equation remains valid over the set of positive $k$th powers provided the equation has at least $(1+o(1))k$ log $k$ variables. We thus completely describe which diagonal forms are partition regular and which are not, given sufficiently many variables. In addition, we prove a supersaturated version of Rado’s theorem for a linear equation restricted either to squares minus one or to logarithmically-smooth numbers.

中文翻译：

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Rado关于平方和更高次方的判据

我们在五个或更多变量中建立了广义勾股定律的分配规律。此外，我们展示了Rado的分区正则方程的刻画如何在$ k $次幂的集合上仍然有效，前提是该方程至少具有$（1 + o（1））k $ log $ k $变量。因此，在给定足够多的变量的情况下，我们完全描述了哪些对角线形式是分区规则的而哪些不是。此外，我们证明了线性方程的Rado定理的超饱和版本，该线性方程限制为平方减去1或对数平滑数。