# An infinite version of the Dilworth theorem

The Dilworth theorem for finite posets implies that a finite poset contains either a "large" chain or a "large" antichain. I am sure I saw an infinite version of this :

An infinite poset has either an infinite chain or an infinite antichain.

But I can't find a reference to that statement. What is the reference or a proof?

There is an exercise in Stanley's Enumerative Combinatorics (Ex. 12 in Chapter 3 of Vol. 1): "True or false: if every chain and every antichain of a poset $$P$$ is finite, then $$P$$ is finite." It also contains a direct proof in the Solutions section, not using Ramsey theorem.
• I should mention that in Reverse Mathematics, the form of Ramsey's theorem that is needed here is quite weak -- weaker than the statement that for each $f:\mathbb N\to\mathbb N$, range($f$) exists as a set. Oct 30 '19 at 7:18