  We prove that for an arbitrary subtree $T$ of $2^{<\omega}$ with each element
extendable to a path, a given countable class $\mathcal{M}$ closed under
disjoint union, and any set $A$, if none of the members of $\mathcal{M}$
strongly $k$-enumerate $T$ for any $k$, then there exists an infinite set
contained in either $A$ or $\bar{A}$ such that for every $C\in\mathcal{M}$,
$C\oplus G$ also does not strongly $k$-enumerate $T$. We give applications of
this result, which include: (1) $\mathsf{RT_2^2}$ doesn't imply
$\mathsf{WWKL_0}$; (2) (Ambos-Spies et al.2004) $\mathsf{DNR}$ is strictly
weaker than $\mathsf{WWKL_0}$; (3) (Kjos-Hanssen 2009) for any Martin-L\"{o}f
random set $A$ either $A$ or $\bar{A}$ contains an infinite subset that does
not compute any Martin-L\"{o}f random set; etc. We also discuss further
generalizations of this result.
