Sets without k‐term progressions can have many shorter progressions. Issue 3 (15th December 2020) - Fox, Jacob
Abstract: 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.