A problem in prime number theory

Question

I was wondering if anybody here might provide me with a hint for this rather innocuous-looking problem:

If $X:= \{pq: p, q \mbox{ are prime numbers and } p\neq q\}.$ In addition, let us suppose that $A\subseteq X$ and $B=X\setminus A.$ Prove that there is an infinite set $P$ of prime numbers such that $Y:= \{pq: p, q \in P \mbox{ and } p \neq q\}$ is contained in $A$ or $B$.

Thank you!

Answer 1

This is an application of Ramsey's theorem. One way to state the relevant version in the language of graphs is:

Suppose an infinite set $X$ of vertices is given, and each edge between two distinct vertices is colored either red or blue. Then there is an infinite subset $H$ of vertices with all edges between them of the same color.

Apply the result here with the set of prime numbers playing the role of $X$. Color an edge between $p$ and $q$ red iff $pq\in A$ and blue otherwise (that is, iff $pq\in B$). The infinite $H$ the theorem guarantees has the property that $\{pq\mid p,q\in H,p\ne q\}$ is a subset of $A$ or of $B$; the first case corresponds to edges between vertices in $H$ being red, the second corresponds to them being blue.