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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?

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).

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  • $\begingroup$ "infinite-dimensional" means (infinite countable)-dimensional? it was proved some time ago that there are several infinite cardinals :) $\endgroup$
    – YCor
    Commented Apr 27, 2020 at 10:35
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    $\begingroup$ Yes: for concreteness, define $\mathbb{F}_q \mathbb{P}^{\infty}$ to be the union of the sequence $\mathbb{F}_q \mathbb{P}^1 \subseteq \mathbb{F}_q \mathbb{P}^2 \subseteq \mathbb{F}_q \mathbb{P}^3 \subseteq \cdots$. $\endgroup$
    – Adam P. Goucher
    Commented Apr 27, 2020 at 11:09

1 Answer 1

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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).

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