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 will show that this theorem is 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 and others on the reverse math of Ramsey's theorem for pairs.

翻译：假设你有一个不可计算集合 $X$，并且你想找到一个集合 $A$，使得它的所有无限子集都能计算 $X$。有几种方法可以实现这一点，但它们似乎都产生一个相当稀疏的集合 $A$。我们证明这在以下技术意义上是必要的：如果 $X$ 不可计算，且 $A$ 是一个具有正下密度的集合，那么 $A$ 存在一个无限子集无法计算 $X$。我们将证明该定理在某些意义上是精确的，并给出一个用柯尔莫哥洛夫复杂度表述的量化版本。我们的结果使用了修正的Mathias力迫方法，并基于Seetapun等人在成对拉姆齐定理逆数学方面的工作。