I want some results for the following statement.
Let $S_{1}$, $S_{2}$, and $S_{3}$ be set of primes. There are infinitely many pairs of two integers $\left(m,n\right)$ such that (1) any prime divisor of $m$ is in $S_{1}$, (2) any prime divisor of $b$ is in $S_{2}$, and (3) any prime divisor of $m+n$ is in $S_{3}$.
You may give several conditions for $S_{i}$. For example, $S_{1}=S_{2}$, each $S_{i}$ is infinite, etc.
What are the suitable keywords to search problems like this statement?