Ramsey's theorem
Relationship to the axiom of choice

In reverse mathematics, there is a significant difference in proof strength between the version of Ramsey's theorem for infinite graphs (the case n = 2) and for infinite multigraphs (the case n ≥ 3). The multigraph version of the theorem is equivalent in strength to the arithmetical comprehension axiom, making it part of the subsystem ACA of second-order arithmetic, one of the big five subsystems in reverse mathematics. In contrast, by a theorem of David Seetapun, the graph version of the theorem is weaker than ACA, and (combining Seetapun's result with others) it does not fall into one of the big five subsystems. Over ZF, however, the graph version implies the classical Kőnig's lemma, whereas the converse implication does not hold, since Kőnig's lemma is equivalent to countable choice from finite sets in this setting.
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