Fix a positive integer $k\geq 2$. Can we always find infinitely many primes $p_1,p_2,\ldots$ such that the sum of any $k$ of them is composite?
If $k$ is even, it is clear that we can. We only need to not choose $2$, so that all the chosen primes are odd, and the sum of any $k$ of them is even, hence composite. But what about if $k$ is odd?