Coding information into all infinite subsets of a dense set
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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.
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