A finite or infinite matrix A is image partition regular provided that whenever $${\mathbb {N}}$$ N is finitely colored, there must be some $$\vec {x}$$ x → with entries from $${\mathbb {N}}$$ N such that all entries of $$A \vec {x}$$ A x → are in the same color class. Comparing to the finite case, infinite image partition regular matrices seem more harder to analyze. The concept of centrally image partition regular matrices were introduced to extend the results of finite image partition regular matrices to infinite one. In this paper, we shall introduce the notion of C-image partition regular matrices, an interesting subclass of centrally image partition regular matrices. Also we shall see that many of known centrally image partition regular matrices are C-image partition regular.

中文翻译：

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无限图像分区正则矩阵 - C-sets 中的解决方案

有限或无限矩阵 A 是图像分区规则，只要 $${\mathbb {N}}$$ N 是有限着色的，必须有一些 $$\vec {x}$$ x → 来自 $${ \mathbb {N}}$$ N 使得 $$A \vec {x}$$ A x → 的所有条目都在相同的颜色类别中。与有限情况相比，无限图像分区正则矩阵似乎更难分析。引入了中心图像分割正则矩阵的概念，将有限图像分割正则矩阵的结果扩展到无穷大。在本文中，我们将介绍 C 图像分区正则矩阵的概念，这是集中图像分区正则矩阵的一个有趣的子类。我们还将看到许多已知的中央图像分区正则矩阵是 C 图像分区正则矩阵。