I've been given this as an exercise.
If P is a partially ordered infinite space, there exists an infinite subset S of P that is either chain or antichain.
This exercise was given in the Axiom of Choice section of the class. I answered it using Zorn's lemma but I'm not 100% sure. Can you give me any hints?
I used Hausdorff's maximal principle and said that there would be a chain that is maximal. And I used Zorn's lemma on the partially ordered subspace of P that contains all the subsets of P that are antichains. But I don't know if these solutions are correct and even if they are I'm stuck at the infinite part.
I was informed that this question is missing context or other details. I'm sorry but that was exactly how it was given to me so I don't know how to correct it.
HINT: The easiest argument is to use the infinite Ramsey theorem; you need just two colors, one for pairs that are related in $P$, and one for pairs that are incomparable in $P$. There is a fairly easy proof of the theorem at the link.