A note on edge colorings and trees

Pub Date : 2022-08-20 DOI:10.1002/malq.202100019

Adi Jarden, Ziv Shami

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Abstract

We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal κ has a homogeneous set of size κ provided that the number of colors μ satisfies $μ^{+} < κ$ . Another result is that an uncountable cardinal κ is weakly compact if and only if κ is regular, has the tree property, and for each $λ, μ < κ$ there exists $κ^{*} < κ$ such that every tree of height μ with λ nodes has less than $κ^{*}$ branches.

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关于边缘着色和树木的注释

指出了某些边着色的齐次集的存在性与某些树的分支的存在性之间的联系。因此，我们得到了任何基数κ的局部加性着色(本文引入的一个概念)都有一个大小为κ的齐次集，只要颜色的个数μ满足μ + ＆lt;κ $\mu ^+<\kappa$。另一个结果是不可数基数κ是弱紧的，当且仅当κ是正则的，具有树的性质，并且对于每个λ， μ ＆lt;κ $\lambda ,\mu <\kappa$存在κ * ＆lt;κ $\kappa ^*<\kappa$使得每棵高度为μ且节点为λ的树都有小于κ∗$\kappa ^*$的分支。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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