$A,B,C$ can be anything (e.g. I proved it for Galois extensions of $\Bbb Q$). Can someone find a proof or a counterexample (for any class of $A,B,C$, e.g. groups, rings) that if $A\subseteq C$ then $AB\cap C\subseteq A(B\cap C)$? (since the $\supseteq$ is obvious). My proof is really really lo...

I read in the english Ultrafilter article on Wikipedia that one can prove that a non-principal ultrafilter $D$ on $\omega$ is a Ramsey ultrafilter if and only if for every coloration $c:[\omega]^2 \longrightarrow 2 $ there exists an element of $D$ that is homogeneous in color, and I've been tryin...