Let $p_n$ be the $n$-th prime number, as usual:
$p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc.
For $k=1,2,3,\ldots$, define
$$
g_k = \liminf_{n \rightarrow \infty} (p_{n+k} – p_n).
$$
Thus the twin prime conjecture asserts $g_1 = 2$.
Zhang's theorem (= weak twin prime conjecture) asserts $g_1 < \infty$.
The prime $m$-tuple conjecture asserts
$g_2 = 6$ (infinitely many prime triplets),
$g_3 = 8$ (infinitely many prime quadruplets), "etcetera" (with $m=k+1$).
Can Zhang's method be adapted or extended to prove $g_k < \infty$
for any (all) $k>1$?
Added a day later: Thanks for all the informative comments and answers!
To summarize and update (I hope I'm attributing correctly):
0) [Eric Naslund] The question was already raised in the
Goldston-Pintz-Yıldırım
paper. See Question 3 on page 3:
Assuming the Elliott-Halberstam conjecture, can it be proved that
there are three or more primes in admissible $k$-tuples with large enough $k$?
Even under the strongest assumptions, our method fails to prove
anything about more than two primes in a given tuple.
1) [several respondents]
As things stand now, it does not seem that Zhang's technique or any
other known method can prove finiteness of $g_k$ for $k > 1$.
The main novelty of Zhang's proof is a cleverly weakened estimate
a la Elliott-Halberstam, which is well short of "the strongest assumptions"
mentioned by G-P-Y.
2) [GH] For $k>1$, the state of the art remains for now as it was pre-Zhang,
giving nontrivial bounds not on $g_k$ but on
$$
\Delta_k := \liminf_{n \rightarrow \infty} \frac{p_{n+k} – p_n}{\log n}.
$$
The Prime Number Theorem (or even Čebyšev's technique) trivially yields
$\Delta_k \leq k$ for all $k$; anything less than that is nontrivial.
Bombieri and Davenport obtained $\Delta_k \leq k – \frac12$;
the current record
is $\Delta_k \leq e^{-\gamma} (k^{1/2}-1)^2$.
This is positive for $k>1$ (though quite small for $k=2$ and $k=3$,
at about $0.1$ and $0.3$), and for $k \rightarrow \infty$ is
asymptotic to $e^{-\gamma} k$ with $e^{-\gamma} \approx 0.56146$.
3) [Nick Gill, David Roberts] Some other relevant links:
Terry Tao's June 3 exposition of Zhang's result
and the work leading up to it;
The "Secret Blogging Seminar" entry and thread that has already
brought the bound on $g_1$ from Zhang's original $7 \cdot 10^7$
down to below $5 \cdot 10^6$;
A PolyMath page
that's keeping track of these improvements with links to the
original arguments, supporting computer code, etc.;
A Polymath proposal
that includes the sub-project of achieving further such improvements.
4) [Johan Andersson] A warning: phrases such as
"large prime tuples in a given [length] interval"
(from the Polymath proposal) refer not to configurations
that we can prove arise in the primes but to admissible
configurations, i.e. patterns of integers that could all be prime
(and should all be prime infinitely often, according to the
generalized prime $m$-tuple [a.k.a. weak Hardy-Littlewood] conjecture,
which we don't seem to be close to proving yet). Despite appearances,
such phrasings do not bear on a proof of $g_k < \infty$ for $k>1$,
at least not yet.
Best Answer
In Goldston, Pintz, and Yildirim paper Primes in tuples I, they show that under the assumption of the Elliott Halberstam Conjecture,
$$\liminf_{n\rightarrow\infty}p_{n+1}-p_n \leq 16$$
and they leave the following question on page 3:
From what I understand, the issue is increasing a coefficient from $1$ to $2$.
Let $\mathcal{H}=\left\{ 1,\dots,h_{k}\right\}$ be our admissible set, and suppose that $\max_{i}h_{i}\leq x.$ The approach is to look at the sum
$$\sum_{x<n\leq 2x}\left(\sum_{i=1}^{k}\vartheta\left(n+h_{i}\right)-\log(3x)\right)W(n),$$
where $\vartheta(n)=1_{\mathcal{P}}(n)\log n$, $1_{\mathcal{P}}(n)$ is the indicator function for the primes, and $W(n)$ is a positive weight function. If this sum is positive, then one of the terms must be positive, so for some $x<n\leq2x$ we have
$$\sum_{i=1}^{k}\vartheta\left(n+h_{i}\right)>\log(3x),$$
and since $\log(n+h_{i})\leq\log(3x)$ for all $n$ in our range, it follows that there are at least two indices $i\neq j$ such that
$$\vartheta(n+h_{i}),\ \vartheta(n+h_{j})\neq0.$$
Selberg advocated that in general for ease of calculation one should take a positive weight function to be a square, $W(n)=\lambda(n)^{2},$ so the goal is to prove the inequality $$\sum_{x<n\leq2x}\sum_{i=1}^{k}\vartheta\left(n+h_{i}\right)\lambda(n)^{2}>\log(3x)\sum_{x<n\leq2x}\lambda(n)^{2}$$
for some choice of $\lambda(n).$ In Goldston, Pintz, and Yildirim's paper, they choose
$$\lambda(n)=\frac{1}{\left(k+l\right)!}\sum_{\begin{array}{c} d|P(n)\\ d\leq R \end{array}}\mu(d)\log\left(\frac{R}{d}\right)^{k+l}$$
where $P(n)=\prod_{j=1}^{k}\left(n+h_{j}\right)$, and $R$ depends on $x$. To use the same approach for $3$ terms, we would need to examine the sum
$$\sum_{x<n\leq2x}\left(\sum_{i=1}^{k}\vartheta\left(n+h_{i}\right)-2\log(3x)\right)\lambda(n)^{2},$$
and show that
$$\sum_{x<n\leq2x}\sum_{i=1}^{k}\vartheta\left(n+h_{i}\right)\lambda(n)^{2}>2\log(3x)\sum_{x<n\leq2x}\lambda(n)^{2},$$
for a suitable choice of $\lambda(n)$. Increasing the coefficient to a $2$ seems to be a fundamental issue, and hopefully an expert can explain why this is the case.