My (current) understanding of the Green-Tao theorem is that it asserts the following:
Theorem (Green-Tao): Let $\{p_i\}_{i \geq 1}$ denote the sequence of primes. Then for any $M \in (0, \infty)$, there exists an integer $j \geq M$ and a finite, strictly increasing sequence $a_1, a_2, \cdots, a_j$ of positive integers such that $p_{a_1}, p_{a_2}, p_{a_3}, \cdots, p_{a_j}$ is an arithmetic progression.
The Green-Tao theorem is often very simply stated as "the primes contain arbitrarily long arithmetic progressions", and this assertion was, for me, slightly ambiguous. Initially, I interpreted as follows.
Theorem (Green-Tao, misinterpreted by me): Let $\{p_i\}_{i \geq 1}$ denote the sequence of primes. Then for any $M \in (0, \infty)$, there exists an integer $j \geq M$ and an integer $k \in \mathbb{N}$ such that $p_{k}, p_{k+1}, \cdots, p_{k+j-1}$ is an arithmetic progression.
Anyways, my questions are the following. First, it would be great if you could verify that my original statement of Green-Tao above is correct, and I don't have any of this mixed up. Second, what can we say about the second, misinterpreted statement? Is it true (perhaps a corollary somehow of the first, correct statement)? False? Or is it open?