This is Part 2 in a series of papers about the growth of regular partitions in hereditary properties $3$-uniform hypergraphs. The focus of this paper is the notion of weak hypergraph regularity, first developed by Chung, Chung-Graham, and Haviland-Thomason. Given a hereditary property of $3$-uniform hypergraphs $\mathcal{H}$, we define a function $M_{\mathcal{H}}:(0,1)\rightarrow \mathbb{N}$ by letting $M_{\mathcal{H}}(ε)$ be the smallest integer $M$ such that all sufficiently large elements of $\mathcal{H}$ admit weak regular partitions of size at most $M$. We show the asymptotic growth rate of such a function falls into one of four categories: constant, polynomial, between single and double exponentials, or tower. These results are a crucial component in Part 3 of the series, which considers vertex partitions associated to a stronger notion of hypergraph regularity.
Partition regularity and multiplicatively syndetic sets
我们展示了如何乘法syndetic集可以用于研究的伸缩不变系统的多项式方程的分割正则性。特别地，我们证明了一个膨胀不变的多项式方程组是划分正则的当且仅当它在每个乘法合集内都有解。我们还采用了Green-Tao和Chow-Lindqvist-Prendiville的方法来开发Roth密度增量策略的联合版本。然后，该参数用于获得形式为$\{x, d, x + d, x + 2d\}$的配置的Rado数的界限。
Partition regularity in imaginary quadratic rings of integers
Universidad de Chile（智利大学）；Centro de Modelamiento Matemático（数学建模中心）；Complutense University of Madrid（马德里康普顿斯大学）；Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学）；Virginia Tech（弗吉尼亚理工大学）
Uniformity of multiplicative functions and partition regularity of some quadratic equations
自Schur和van der Waerden定理以来，人们对线性方程证明了大量的剖分正则性结果，但对非线性方程的剖分正则性研究进展甚微，最困难的情况是三元方程。我们证明分区正则性的某些方程涉及形式在三个变量，例如，显示的方程$16x^2+9y^2=n^2$和$x^2+y^2-xy=n^2$是分区规则，其中$n$允许自由变化的$\mathbb{N}$。对于每一个这样的问题，我们建立一个密度模拟，可以制定在遍历条款作为一个递归属性的行动由膨胀的概率空间。我们的关键工具，建立这样的递归性质是一个分解结果的乘性函数，这是独立的利益。粗略地说，它指出模为1的任意乘性函数可以分解为两项，一项是近似周期性的，另一项具有三次的小Gowers一致性范数。
