Lines imply spaces in density Ramsey theory (Q788721)

\textit{E. Szemerédi} [Acta. Arith. 27, 199--245 (1975; Zbl 0303.10056)] showed that van der Waerden's theorem on arithmetic progressions admits a density version. It is an outstanding question whether the partition result on projective and affine \(k\)-dimensional spaces over a finite field \(\mathbb F\) admits a density version. In this nice paper the authors show that if a density version for 1-dimensional affine spaces over \(\mathbb F\) should be valid (which is the case for \(| \mathbb F| =2,3)\) then a density version holds for arbitrary \(k\)-dimensional affine spaces over \(\mathbb F\).