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.

翻译：假设你有一个不可计算集 $X$，并且你想找到一个集合 $A$，使得 $A$ 的所有无穷子集都能计算 $X$。有多种方法可以实现这一点，但所有这些方法似乎都会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 $A$ 存在一个无穷子集不能计算 $X$。我们还证明了关于 PA 度的类似结果：如果 $X$ 不可计算且 $A$ 是一个具有正下密度的集合，那么 $A$ 存在一个无穷子集不具有 PA 度。我们将证明这些定理在某种意义上是最优的，并证明一个用柯尔莫哥洛夫复杂度表述的定量版本。我们的结果使用了 Mathias 力迫的改进版本，并基于 Seetapun、Liu 及其他人在对偶 Ramsey 定理逆向数学方面的工作。