KöMaL 2016 A669

Determine whether the set of rational numbers can be ordered to in a sequence $q_1,q_2,\ldots$ in such a way that there is no sequence of indices $1\le i_1<i_2<\dots<i_6$ such that $q_{i_1},q_{i_2},\ldots,q_{i_6}$ form an arithmetic progression.