Define the length of a set of arithmetic progressions of natural numbers $A=\lbrace A_1, A_2, \ldots \rbrace$ to be $\min_i | A_i |$: the length of the shortest sequence among all the progressions. ...
Van der Corput  proved that there are infinitely many arithmetic progressions of primes of length 3 (PAP-3). (Green & Tao  famously extended this theorem to length $k$.) But taking this in ...