Random-set Szemerédi threshold conjecture

For $n\in\mathbb{N}$, let $[n]_p$ denote the random subset of $[n]=\{1,\ldots,n\}$ obtained by selecting each integer independently with probability $p$. A set $I$ is $(\delta,k)$-Szemerédi if every subset of $I$ of cardinality at least $\delta|I|$ contains an arithmetic progression of length $k$.

Random-set Szemerédi threshold conjecture. For every $\delta>0$ and every integer $k\geq 3$, there exist positive constants $c$ and $C$ such that

$$\lim_{n\to\infty}\mathbb{P}\bigl([n]_p\text{ is $(\delta,k)$-Szemerédi}\bigr)=\begin{cases}0,&p<c n^{-1/(k-1)},\\1,&p>C n^{-1/(k-1)}.\end{cases}$$

The conjecture asserts that the natural obstruction gives the sharp threshold, improving substantially on the $n^{-o(1)}$ range obtained from Green–Tao pseudorandomness methods. Its status is open in the supplied text.