Sets without $k$-term progressions can have many shorter progressions

Abstract

Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic progressions in a sequence $a_1<a_2<\dots <a_n$ of $n$ integers which contains no $k$-term arithmetic progression. For all integers $k>s\ge 3$, we prove that $$\underset{n\to \infty }{\lim }\frac{\log f_{s,k}(n)}{\log n}=2,$$ 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.