Let $1\le k<\ell$ be integers and define $F_k(N,\ell )$ to be minimal such that every set $A\subset \mathbb{N}$ of size $N$ which contains at least $F_k(N,\ell )$ many $k$-term arithmetic progressions must contain an $\ell$-term arithmetic progression. Find good upper bounds for $F_k(N,\ell )$. Is it true that$$F_3(N,4)=o(N^2)?$$Is it true that for every $\ell >3$$$\underset{N\to \infty }{\lim }\frac{\log F_3(N,\ell )}{\log N}=2?$$