The statement (**) is inconsistent with ZFC, because [my answer to your previous question](http://mathoverflow.net/a/206954/1946) shows that even the finitary form is inconsistent. There is a coloring of countable ordinals, such that for no ordinals $\alpha,\beta$ do the ordinals $\alpha+\beta\cdot\omega$ and $\alpha+\beta\cdot\omega^2$ get the same color.