I'm trying to prove the following: Every infinite directed graph has an infinite subset of vertices that induces one of the following:
- a strict linear order
- a weak linear order
- an independent set without loops
- loops at each vertex and otherwise no edges
I was thinking of seeing $(V,E)$ as a set with a relation $(P,\leq)$ and since it can be seen as an infinite poset with reachability I can use Ramsey's theorem by coloring every pair as either connected or not, then we have an infinite chain or infinite antichain.
The chain is an infinite path, if it has loops at every vertex then it is reflexive and a weak linear order. If not then it is a strong linear order.
If it has an infinite anti chain, it is possible that the vertices in the antichain have loops aka $v \leq v, \forall v\in A$, $A$ antichain, if not then they are completely independent with $v\not \in X\in {A\choose 2 }$, for all edges $X$.
I'm not sure if these thoughts can be put in a coherent proof.