We consider finite colourings of finite products \(X_1\times X_2\times \cdot \cdot \cdot \times X_n\) of infinite sets and determine what is the minimal number of colours a subproduct \(Y_1\times Y_2\times \cdot \cdot \cdot \times Y_n\) of infinite subsets could achieve.

中文翻译：

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着色有限产品

我们考虑无限集合的有限乘积\(X_1\times X_2\times \cdot \cdot \cdot \times X_n\)的有限着色，并确定子积 \(Y_1\times Y_2\times \cdot \cdot \cdot \times Y_n\)的无限子集可以实现。