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.

中文翻译：

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没有k项级数的集合可以具有许多较短的级数

令f _{s，  k}（n）是一组不包含k项算术级数的n个整数中s项算术级数的最大可能数目。对于所有的固定整数ķ  > 小号 ≥3 ，我们证明了˚F_{小号，  ķ}（Ñ）=  ñ ^{2 -  Ö（1）} ，其答案ERDOS的一个老问题。实际上，我们证明了f _{s，  k}（n）的上下界 这表明其增长与Szemerédi定理的界线密切相关。