Let be any linearly independent polynomials with zero constant term. We show that there exists such that any subset of of size at least contains a nontrivial polynomial progression , provided that the characteristic of is large enough.
"On the polynomial Szemerédi theorem in finite fields." Duke Math. J. 168 (5) 749 - 774, 1 April 2019. https://doi.org/10.1215/00127094-2018-0051