Cone avoiding closed sets

Abstract

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) $\mathsf{R}{\mathsf{T}}_2^2$ doesn't imply $\mathsf{WWK}{\mathsf{L}}_0$; (2) (Ambos-Spies et al.2004) $\mathsf{DNR}$ is strictly weaker than $\mathsf{WWK}{\mathsf{L}}_0$; (3) (Kjos-Hanssen 2009) for any Martin-Löf random set $A$ either $A$ or $\bar{A}$ contains an infinite subset that does not compute any Martin-Löf random set; etc. We also discuss further generalizations of this result.

22 pages