> What stopped Patey and Yokoyama from proving Ramsey's Theorem For Pairs/Triples by saying "for any pair which satisfies some relation X, there exists another

> What stopped Patey and Yokoyama from proving Ramsey's Theorem For Pairs/Triples by saying "for any pair which satisfies some relation X, there exists another pair which also satisfies relation X"? Because it's not an axiom? I feel like I'm not understanding your question completely