On subsets of containing no -term progressions

In this paper we prove that for any fixed integer k and any prime power q ≥ k , there exists a subset of F q 2 k of size q 2 ( k − 1 ) + q k − 1 − 1 which contains no k points on a line, and hence no k -term arithmetic progressions. As a corollary we obtain an asymptotic lower bound as n → ∞ for r k ( F q n ) when q ≥ k , which can be interpreted as the finite field analogue of Behrend’s construction for longer progressions.