A note on maximal progression-free sets

Erd?s et al Greedy algorithm, arithmetic progressions, subset sums and divisibility, Discrete Math. 200 (1999) 119-135.] asked whether there exists a maximal set of positive integers containing no three-term arithmetic progression and such that the difference of its adjacent elements approaches infinity. This note answers the question affirmatively by presenting such a set in which the difference of adjacent elements is strictly increasing. The construction generalizes to arithmetic progressions of any finite length.