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| '''Folkman's theorem''' is a [[theorem]] in mathematics, and more particularly in [[arithmetic combinatorics]] and [[Ramsey theory]]. According to this theorem, whenever the [[natural number]]s are partitioned into finitely many subsets, there exist arbitrarily large sets of numbers all of whose sums belong to the same subset of the partition.<ref name="grs">{{citation|title=Ramsey Theory|first1=Ronald L.|last1=Graham|author1-link=Ronald Graham|first2=Bruce L.|last2=Rothschild|author2-link=Bruce Lee Rothschild|first3=Joel H.|last3=Spencer|author3-link=Joel Spencer|publisher=Wiley-Interscience|year=1980|contribution=3.4 Finite sums and finite unions (Folkman's theorem)|pages=65–69}}.</ref> The theorem had been discovered and proved independently by several mathematicians,<ref name="Rado">{{citation
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| | last = Rado | first = R. | authorlink = Richard Rado
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| | contribution = Some partition theorems
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| | mr=297585
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| | location = Amsterdam
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| | pages = 929–936
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| | publisher = North-Holland
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| | title = Combinatorial theory and its applications, III: Proc. Colloq., Balatonfüred, 1969
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| | year = 1970}}.</ref><ref name="Sanders">{{citation
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| | last = Sanders | first = Jon Henry
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| |mr = 2617864 | |
| | publisher = Yale University
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| | series = Ph.D. thesis
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| | title = A generalization of Schur's theorem
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| | year = 1968}}.</ref> before it was named "Folkman's theorem", as a memorial to [[Jon Folkman]], by [[Ronald Graham|Graham]], [[Bruce Lee Rothschild|Rothschild]], and [[Joel Spencer|Spencer]].<ref name="grs"/>
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| ==Statement of the theorem==
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| Let '''N''' be the set {1, 2, 3, ...} of positive integers, and suppose that '''N''' is partitioned into ''k'' different subsets ''N''<sub>1</sub>, ''N''<sub>2</sub>, ... ''N''<sub>''k''</sub>, where ''k'' is any positive integer. Then Folkman's theorem states that, for every positive integer ''m'', there exists a set ''S''<sub>''m''</sub> and an index ''i''<sub>''m''</sub> such that ''S''<sub>''m''</sub> has ''m'' elements and such that every sum of a nonempty subset of ''S''<sub>''m''</sub> belongs to ''N''<sub>''i''<sub>''m''</sub></sub>.<ref name="grs"/>
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| ==Relation to Rado's theorem and Schur's theorem==
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| [[Schur's theorem]] in Ramsey theory states that, for any finite partition of the positive integers, there exist three numbers ''x'', ''y'', and ''x'' + ''y'' that all belong to the same partition set. That is, it is the special case ''m'' = 2 of Folkman's theorem.
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| [[Rado's theorem (Ramsey theory)|Rado's theorem]] in Ramsey theory concerns a similar problem statement in which the integers are partitioned into finitely many subsets; the theorem characterizes the integer matrices '''A''' with the property that the [[system of linear equations]] {{nowrap|1='''A''' ''x'' = 0}} can be guaranteed to have a solution in which every coordinate of the solution vector ''x'' belongs to the same subset of the partition. A system of equations is said to be ''regular'' whenever it satisfies the conditions of Rado's theorem; Folkman's theorem is equivalent to the regularity of the system of equations
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| :<math>x_T = \sum_{i\in T}x_{\{i\}},</math> | |
| where ''T'' ranges over each nonempty subset of the set {{nowrap|{1, 2, ..., ''m''}.}}<ref name="grs"/>
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| ==Multiplication versus addition==
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| It is possible to replace addition by multiplication in Folkman's theorem: if the natural numbers are finitely partitioned, there exist arbitrarily large sets ''S'' such that all products of nonempty subsets of ''S'' belong to a single partition set. Indeed, if one restricts ''S'' to consist only of [[power of two|powers of two]], then this result follows immediately from the additive version of Folkman's theorem. However, it is open whether there exist arbitrarily large sets such that all sums and all products of nonempty subsets belong to a single partition set. It is not even known whether there must necessarily exist a set of the form {{nowrap|{''x'', ''y'', ''x'' + ''y'', ''xy''}}} for which all four elements belong to the same partition set.<ref name="grs"/>
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| ==Previous results==
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| Variants of Folkman's theorem had been proved by [[Richard Rado]] and by J. H. Sanders.<ref name="Rado"/><ref name="Sanders"/><ref name=" grs"/> Folkman's theorem was named in memory of [[Jon Folkman]] by [[Ronald Graham]], [[Bruce Lee Rothschild]], and [[Joel Spencer]], in their book on [[Ramsey theory]].<ref name="grs"/>
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| ==References==
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| {{reflist}}
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| [[Category:Theorems in discrete mathematics]]
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| [[Category:Ramsey theory]]
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| [[Category:Sumsets]]
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| [[Category:Additive combinatorics]]
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| [[Category:Additive number theory]]
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| [[Category:Theorems in combinatorics]]
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