Let $\mathbb{F}\_q$ be a finite field and $k$ be a positive integer. If we colour each point of the infinite-dimensional projective space $\mathbb{F}\_q \mathbb{P}^{\infty}$ with one of $k$ colours, can we necessarily find an infinite-dimensional monochromatic projective subspace?




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There are a couple of observations that hint that the answer is 'yes':


**Observation 1:** The statement for $\mathbb{F}\_2$ is true, and is equivalent to Hindman's theorem. The reason for asking this question is that it would provide an interesting generalisation of Hindman's theorem.


**Observation 2:** If we weaken the problem to finding an $n$-dimensional monochromatic projective subspace (where $n$ is finite), the statement is also true (and follows, for example, from the 'Vector Space Ramsey Theorem' in the paper *Ramsey's Theorem for Spaces* by Joel H. Spencer).



Ramsey theory in infinite-dimensional projective spaces
After further investigation, it appears that disappointingly the answer is 'no', and a proof appears in Lemma 2.4 of *Partition Theorems for Subspaces of Vector Spaces* (Cates and Hindman, 1975).


