The Green–Tao conjecture for affine spaces of irreducible polynomials

Let $\mathbf{F}$ be a finite field, and let $\mathbf{F}[t]$ denote its polynomial ring. An affine space in $\mathbf{F}[t]$ is a set of the form

$$\{f+Pg: \deg(P)<k\},$$

where $f,g\in\mathbf{F}[t]$ and $g\ne 0$.

Green–Tao conjecture for function fields. For every finite field $\mathbf{F}$, the monic irreducible polynomials in $\mathbf{F}[t]$ contain affine spaces of arbitrarily high dimension.

This is the function-field analogue of the Green–Tao theorem that the primes contain arithmetic progressions of arbitrarily large length. In the formulation given here, the claim concerns configurations of monic irreducible polynomials parametrized by polynomials of increasing degree.