  Suppose you have an uncomputable set $X$ and you want to find a set $A$, all
of whose infinite subsets compute $X$. There are several ways to do this, but
all of them seem to produce a set $A$ which is fairly sparse. We show that this
is necessary in the following technical sense: if $X$ is uncomputable and $A$
is a set of positive lower density then $A$ has an infinite subset which does
not compute $X$. We also prove an analogous result for PA degree: if $X$ is
uncomputable and $A$ is a set of positive lower density then $A$ has an
infinite subset which is not of PA degree. We will show that these theorems are
sharp in certain senses and also prove a quantitative version formulated in
terms of Kolmogorov complexity. Our results use a modified version of Mathias
forcing and build on work by Seetapun, Liu, and others on the reverse math of
Ramsey's theorem for pairs.
