A note on strong negative partition relations

关于度量空间中 Kuratowski 划分的注记

A note on Kuratowski partitions in metric spaces

本文证明了当 $κ$ 为不大于 $2^{\aleph_0}$ 的最小实值可测基数时，必然存在基数至多 $2^κ$ 且容许 Kuratowski 划分的完备度量空间。

A note on infinite partitions of free products of Boolean algebras

If $A$ is an infinite Boolean algebra the cardinal invariant $\mathfrak{a}(A)$ is defined as the smallest size of an infinite partition of $A$. The cardinal $\mathfrak{a}(A\oplus B)$, where $A\oplus B$ is the free product of the Boolean algebras $A$ and $B$ (whose dual topological space is the product of the dual topological spaces of $A$ and $B$), is below both $\mathfrak{a}(A)$ and $\mathfrak{a}(B)$. The equality $\mathfrak{a}(A\oplus B)=\min\lbrace\mathfrak{a}(A),\mathfrak{a}(B)\rbrace$ is not known to hold for all infinite Boolean algebras $A$ and $B$. Here some lower bounds of $\mathfrak{…

Ramsey theory over partitions II: Negative Ramsey relations and pump-up theorems

In this series of papers we advance Ramsey theory of colorings 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 $Pr_1(\ldots)$ over a partition may be improved to witness $Pr_0(\ldots)$. Since the classical theory corresponds…

A note on the Canonical Ramsey Theorem and Ramsey ultrafilters

We give a characterizations of Ramsey ultrafilters on $\mathscr P(ω)$ in terms of functions $f:ω^n\toω$ and their ultrafilter extensions. To do this, we prove that for any partition $\mathcal P$ of $[ω]^n$ there is a finite partition~$\mathcal Q$ of $[ω]^{2n}$ such that any set $X\subseteq ω$ that is homogeneous for $\mathcal Q$ is a finite union of sets that are canonical for $\mathcal P$.

关于测度论domatic划分的一个注记

A note on measure-theoretic domatic partitions

本文证明了在标准概率空间上，每个保测度的$\aleph_0$-正则Borel图均存在$\mu$-可测顶点$\aleph_0$-染色，使得每个顶点在其邻域中可见所有颜色。

A note on highly connected and well-connected Ramsey theory

We study a pair of weakenings of the classical partition relation $ν\rightarrow (μ)^2_λ$ recently introduced by Bergfalk-Hrušák-Shelah and Bergfalk, respectively. Given an edge-coloring of the complete graph on $ν$-many vertices, these weakenings assert the existence of monochromatic subgraphs exhibiting high degrees of connectedness rather than the existence of complete monochromatic subgraphs asserted by the classical relations. As a result, versions of these weakenings can consistently hold at accessible cardinals where their classical analogues would necessarily fail. We prove some complem…