While reading the posts on ultrafilters by Terry Tao [1,2], I find the following result interesting:
Hindman’s theorem (a special case). Suppose we color the natural numbers using k colors. Then there exists a color c and an infinite set S, all colored with c, such that every finite sum of over S has color c as well.
Also, this theorem is claimed to be rather unpleasant to prove if we insist on not using ultrafilters.
Finite Playground 