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$X$，并且想要找到一个集合$A$，使得$A$的所有无限子集都能计算$X$。有几种方法可以实现这一点，但所有这些方法似乎都产生一个相当稀疏的集合$A$。我们表明，这在以下技术意义上是必要的：如果$X$不可计算且$A$是一个正下密度集合，则$A$存在一个不计算$X$的无限子集。我们还证明了关于PA度的类似结果：如果$X$不可计算且$A$是一个正下密度集合，则$A$存在一个非PA度的无限子集。我们将证明这些定理在某种意义上是最优的，并基于柯尔莫哥洛夫复杂度给出一个定量版本。我们的结果使用了修正的Mathias力迫方法，并建立在Seetapun、Liu等人在对拉姆齐定理的逆向数学研究中关于成对结果的工作之上。