Bergelson–Host–Kra conjecture on monochromatic arithmetic progressions

Let $0<\alpha<1$, let $\epsilon>0$, and let $k=3$ or $4$. For a positive integer $N$, write $[N]=\{1,2,\dots,N\}$. A length-$k$ arithmetic progression in $[N]$ has a common difference $d\neq 0$. Bergelson–Host–Kra conjecture. There exists an $N_0$ such that for all $N\geq N_0$ and every $A\subset[N]$ with $|A|\geq\alpha N$, there is a $d\neq 0$ for which $A$ contains at least $(\alpha^k-\epsilon)N$ distinct arithmetic progressions of length $k$ and common difference $d$. The bound is best possible by the random-set heuristic. The conjecture is now a theorem: Green proved the case $k=3$, and Green and Tao proved the case $k=4$ as well as recovering the case $k=3$; Bergelson, Host and Kra gave a counterexample for $k=5$.