  We prove a theorem ensuring that the compositions of certain Ramsey families
are still Ramsey. As an application, we show that in any finite coloring of
$\mathbb{N}$ there is an infinite set $A$ and an as large as desired finite set
$B$ with $(A+B)\cup (AB)$ monochromatic, answering a question from a recent
paper of Kra, Moreira, Richter, and Robertson. In fact, we prove an iterated
version of this result that also generalizes a Ramsey theorem of Bergelson and
Moreira that was previously only known to hold for fields. Our main new
technique is an extension of the color focusing method that involves trees
rather than sequences.
