We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal characteristics of the continuum. We build on work by Erd̋s, Garti, Jones, Orr, Rado, Shelah and Szemerédi. In particular, we show that a theorem of Jones extends from the natural numbers to the rational ones, but consistently extends only to three further equimorphism classes of countable orderings. This is made possible by applying a 13-year-old theorem of Orr about embedding a given order into a sum of finite orders indexed over the given order.

中文翻译：

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订单类型的极化分区关系

我们将具有两个有序因子的产品划分为两类，其中两个因子都是可数的或有序的，并且至少其中一个是可数的。这将这些产品的分配特性与连续体的基本特性相关联。我们以Erd̋s，Garti，Jones，Orr，Rado，Shelah和Szemerédi的作品为基础。特别地，我们证明了琼斯定理从自然数扩展到有理数，但始终仅扩展到可数序的另外三个同构类。通过应用Orr已有13年历史的定理，该定理可以将给定订单嵌入到在给定订单上建立索引的有限阶总和中。