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.
Categories: math.LO