Sets without k-term progressions can have many shorter progressions

Let f_(s, k)(n) be the maximum possible number of s‐term arithmetic progressions in a set of n integers which contains no k‐term arithmetic progression. For all fixed integers k > s ≥ 3, we prove that f_(s, k)(n) = n^(2 − o(1)), which answers an old question of Erdős. In fact, we prove upper and lower bounds for f_(s, k)(n) which show that its growth is closely related to the bounds in Szemerédi's theorem