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 satisfies (Formula presented.). 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 (Formula presented.) there exists (Formula presented.) such that every tree of height μ with λ nodes has less than (Formula presented.) branches.