A Note on Edge Colorings and Trees

TL;DR

This paper explores the relationship between edge colorings and trees, establishing conditions under which homogeneous sets exist and characterizing weakly compact cardinals through tree properties.

Contribution

It introduces the notion of locally additive colorings and links their homogeneous sets to tree structures, providing new characterizations of weakly compact cardinals.

Findings

Locally additive colorings of a cardinal have homogeneous sets of size equal to the cardinal under certain conditions.

Weakly compact cardinals are characterized by regularity, the tree property, and specific branch conditions in trees.

Connections between edge colorings and tree structures are established, enriching the understanding of large cardinal properties.

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 $\kappa$ has a homogeneous set of size $\kappa$ provided that the number of colors, $\mu$ satisfies $\mu^+<\kappa$. Another result is that an uncountable cardinal $\kappa$ is weakly compact if and only if $\kappa$ is regular, has the tree property and for each $\lambda,\mu<\kappa$ there exists $\kappa^*<\kappa$ such that every tree of height $\mu$ with $\lambda$ nodes has less than $\kappa^*$ branches.