A note on edge colorings and trees

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 satisfiesμ + < κ $\mu ^+<\kappa$ . Another result is that an uncountable cardinal κ is weakly compact if and only if κ is regular, has the tree property, and for eachλ , μ < κ $\lambda ,\mu <\kappa$ there existsκ ∗ < κ $\kappa ^*<\kappa$ such that every tree of height μ with λ nodes has less thanκ ∗ $\kappa ^*$  branches.