In this series of papers we advance Ramsey theory over partitions. In this part, we concentrate on anti-Ramsey relations, or, as they are better known, strong colorings, and in particular solve two problems from [CKS21].

It is shown that for every infinite cardinal λ, a strong coloring on λ⁺ by λ colors over a partition can be stretched to one with λ⁺ colors over the same partition. Also, a sufficient condition is given for when a strong coloring witnessing Pr₁(…) over a partition may be improved to witness Pr₀(…).

Since the classical theory corresponds to the special case of a partition with just one cell, the two results generalize pump-up theorems due to Eisworth and Shelah, respectively.

在本系列论文中，我们提出了关于分区的拉姆齐理论。在这一部分中，我们专注于反拉姆齐关系，或者更广为人知的是，强烈的色彩，特别是解决[CKS21]中的两个问题。

结果表明，对于每个无限基数 λ，分区上 λ ⁺和 λ 颜色的强着色可以扩展到同一分区上具有 λ ^{+颜色的强着色。}另外，给出了当分区上的强着色见证Pr _{1 (…)可以改进为见证Pr 0} (…)时的充分条件。

由于经典理论对应于只有一个单元的分区的特殊情况，因此这两个结果分别推广了 Eisworth 和 Shelah 的泵升定理。