Let $P$ be the set of prime numbers and $P_c$ be the set of chen prime numbers.
What is the upper density of $P_c$ with relative to $P$ ? If the answer is $>0$ then the result in Zhou, Binbin (2009). "The Chen primes contain arbitrarily long arithmetic progressions". Acta Arith. 138 (4): 301–315 is a special case of Green - Tao theorem.