  The $\mathsf{SRT}^2_2$ vs.\ $\mathsf{COH}$ problem is a central problem in
computable combinatorics and reverse mathematics, asking whether every Turing
ideal that satisfies the principle $\mathsf{SRT}^2_2$ also satisfies the
principle $\mathsf{COH}$. This paper is a contribution towards further
developing some of the main techniques involved in attacking this problem. We
study several principles related to each of $\mathsf{SRT}^2_2$ and
$\mathsf{COH}$, and prove results that highlight the limits of our current
understanding, but also point to new directions ripe for further exploration.
