Editing Ramsey's theorem (section)

== 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 [[Second-order arithmetic#Arithmetical comprehension|arithmetical comprehension axiom]], making it part of the subsystem ACA<sub>0</sub> of [[second-order arithmetic]], one of the [[Reverse mathematics#The big five subsystems of second-order arithmetic|big five subsystems]] in reverse mathematics. In contrast, by a theorem of [[David Seetapun]], the graph version of the theorem is weaker than ACA<sub>0</sub>, and (combining Seetapun's result with others) it does not fall into one of the big five subsystems.<ref>{{cite book|last=Hirschfeldt|first=Denis R.|title=Slicing the Truth|title-link=Slicing the Truth|publisher=World Scientific|year=2014|series=Lecture Notes Series of the Institute for Mathematical Sciences, National University of Singapore|volume=28}}</ref>  Over [[Zermelo–Fraenkel set theory|ZF]], however, the graph version implies the classical [[Kőnig's lemma]], whereas the converse implication does not hold,<ref>{{Cite journal|last=Blass|first=Andreas|date=September 1977|title=Ramsey's theorem in the hierarchy of choice principles|url=https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/ramseys-theorem-in-the-hierarchy-of-choice-principles/929A11D7F1897B65F0D1378BB4097910|journal=The Journal of Symbolic Logic|language=en|volume=42|issue=3|pages=387–390|doi=10.2307/2272866|jstor=2272866 |issn=1943-5886}}</ref> since [[Kőnig's lemma]] is equivalent to countable choice from finite sets in this setting.<ref>{{Cite journal|last1=Forster|first1=T.E.|last2=Truss|first2=J.K.|date=January 2007|title=Ramsey's theorem and König's Lemma|url=https://link.springer.com/article/10.1007/s00153-006-0025-z|journal=Archive for Mathematical Logic|language=en|volume=46|issue=1|pages=37–42|doi=10.1007/s00153-006-0025-z|issn=1432-0665}}</ref>