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Question: Isn't there an axiom that says "for any real number, there's always a bigger number"? 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"?

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> 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

Because there doesn't have to be another pair that satisfies that relation. Counterexamples are trivial:

Color the pair blue if its elements are 1 and 0. Color the pair red otherwise.

Color the triplet blue if its elements are 1, -1, and 0. Color the triplet red otherwise.

Then I misunderstood the article when it said

> When this is done, RT22 states that there will exist an infinite monochromatic subset: a set consisting of infinitely many numbers, such that all the pairs they make with all other numbers are the same color.

I read this as saying "There exists and infinite number of x's which satisfy the relation f(x, y) = blue for all values of y over some arbitrary function f()". What am I missing?

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