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