Composite Ramsey theorems via trees

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, addressing a problem of Kra, Moreira, Richter, and Robertson.

In fact, we prove an iterated version of this result, which ensures the existence of monochromatic patterns such as $\{a\circ_1 (b \circ_2 c): \circ_i\in \{+,\cdot\}\}, $ generalizing a Ramsey theorem of Bergelson and Moreira that was previously only known to hold for colorings of $\mathbb{Q}$ rather than colorings of $\mathbb{N}$.

Our main new technique is an extension of the color focusing method that involves trees rather than sequences.