# Generalizations of Chen's theorem

The two famous theorems of Jingrun Chen, both with similar proofs, state (respectively) that all sufficiently large even numbers are the sum of a prime and an element of $P_2$, and that there are infinitely many numbers $n$ such that $\{n, n+2\}$ consists of a prime and an element of $P_2$. Here $P_2$ is the set of numbers with at most two (not necessarily distinct) prime factors.

His latter theorem can be restated as the polynomial $f(n) = n(n+2)$ hits $P_3$ infinitely often. Note in particular that $3 = \deg(f) + 1$, which seems to be the best possible result of this type (any smaller and we run into the infamous parity problem in sieve theory).

Has Chen's theorem been generalized to higher degree polynomials? In particular, let $\mathcal{H}$ be an admissible set in the sense defined in Goldston-Pintz-Yildirim, that for all primes $p$, $\mathcal{H}$ does not contain a complete residue system modulo $p$. Let $|\mathcal{H}| = k$ and let $f(n) = \prod_{h \in \mathcal{H}} (n + h)$. Then can one prove that $f(n)$ hits $P_{k+1}$ infinitely often?

Note: the recent series of results regarding bounded gaps between primes essentially boils down to showing that for any admissible set $\mathcal{H}$, we have $f(n)$ hits $P_{k+l}$ infinitely often for some small positive $l$ (say $l \leq\sqrt{k}$). My question is are there any specific cases where we can take $l = 1$.

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Consider the case $k=3$. If the product of an admissible $3$-tuple had at most $4$ prime factors infinitely often, this would imply that the 3-tuple contains $2$ primes infinity often. In turn, this would imply that there are prime gaps of at most size at most $6$ infinity often.
[Also a correction to your question: one needs $l < k$ to get bounded gaps, some positive $l$ is not sufficient.]