Folkman's theorem

Folkman's theorem is a theorem in mathematics, and more particularly in arithmetic combinatorics and Ramsey theory. According to this theorem, whenever the natural numbers 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. The theorem had been discovered and proved independently by several mathematicians, before it was named "Folkman's theorem", as a memorial to Jon Folkman, by Graham, Rothschild, and Spencer.

Let N be the set {1, 2, 3, ...} of positive integers, and suppose that N is partitioned into k different subsets N1, N2, ... Nk, where k is any positive integer. Then Folkman's theorem states that, for every positive integer m, there exists a set Sm and an index im such that Sm has m elements and such that every sum of a nonempty subset of Sm belongs to Nim.

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.

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 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

where T ranges over each nonempty subset of the set {1, 2, ..., m}.

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 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 {x, y, x + y, xy} for which all four elements belong to the same partition set.

This page was last edited on 21 February 2018, at 10:29.
Reference: https://en.wikipedia.org/wiki/Folkman%27s_theorem under CC BY-SA license.

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