More on maximal line-free sets in Fₚⁿ

For a prime p we construct a subset of Fₚ(k²-k)/2 of size p(k²-k)/2-1 that does not contain progressions of length k. More generally, we show that for any prime power q there is a subset of Fq(k²-k)/2 of size q(k²-k)/2-1 that does not contain k points on a line. This yields the first asympotic lower bounds cⁿ for the size of p-progression-free sets in Fₚⁿ with $c=p-o(1)$, as p tends to infinity.