A Note on Edge Colorings and Trees

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 ${\kappa }^{\ast }<\kappa$ such that every tree of height $\mu$ with $\lambda$ nodes has less than ${\kappa }^{\ast }$ branches.