Many properties of the set of the primes are inherited by any dense subset thereof, such as the existence of solutions to various linear equations. But there’s one property, recently proved, namely the existence of infinitely many gaps p_{n+1}-p_n\leq C bounded by a constant, which is not inherited by dense subsets of the primes. For instance, it is possible to remove basically one half of the primes and be left with a subset of primes \{p'_n\mid n\in\mathbb{N}\} such that p'_{n+1}-p'_n\ll \log p_n.

So instead one could ask if when coloring the primes in finitely many colours, one is guaranteed to find a colour class containing infinitely many pairs of primes whose difference is bounded by a common constant. This does not follow immediately from pigeonhole principle, which only says that one colour class contains infinitely many primes.

However, this follows easily if we recall that the work of Maynard does not only give infinitely many bounded gaps, that is \liminf p_{n+1}-p_n<+\infty, but also for any m that \liminf p_{n+m}-p_n < +\infty. Let H(m) be this liminf.

In other words there are infinitely many (disjoint) intervals of size I_n=[x_n,x_n+H(m)] which contains at least m+1 primes. So if you colour the primes in m colours, and you consider these intervals, you find that each one must contains at least two primes of the same colour. By pigeonhole principle again, there must be a colour for which this occurs infinitely often. Whence the existence in this colour class of infinitely gaps of size at most H(m).