Sum-product patterns in the shifted primes

We show that the set $\mathbb{P}-1$ of shifted primes contains infinitely many sum-product patterns of the form $\{x,x+y,xy\}$ with $x,y$ arbitrarily large distinct integers. More strongly, we can also show that, for any $k\ge 1$, the set $\mathbb{P}-1$ contains longer patterns of the form $\{x,x+y,\dots ,x+ky,xy\}$ with $x,y$ arbitrarily large distinct integers, a statement that contains the Green–Tao theorem as a special case.