TITLE: Does a red/blue coloring of the infinite subsets of $\mathbb{N}$ necessarily give an infinite monochromatic $M\subset \mathbb{N}$?
QUESTION [7 upvotes]: The infinite Ramsey theorem states that for any $n$, if all the subsets of $\mathbb{N}$ of size $n$ are colored red/blue, then there is an infinite $M$ all of whose subsets of size $n$ are monochromatic.
My question is whether there is an analogue if the infinite subsets of $\mathbb{N}$ are coloured red/blue--does this imply that there is an infinite subset $M$ all of whose infinite subsets are monochromatic?

REPLY [2 votes]: It's easy to construct a counterexample using transfinite recursion. Fix a well-ordering of least possible order type of the infinite subsets of $\Bbb{N}$. Now go down this list, at each step making sure that the listed set has both a red subset and a blue subset. Since at each step you have fixed the colour of $< 2^{\aleph_0}$ sets, and each infinite subset of $\Bbb{N}$ has $2^{\aleph_0}$ infinite subsets, there are always plenty of sets to choose from.