IP Set - Hindman's Theorem

Hindman's Theorem

If is an IP set and, then at least one is an IP set. This is known as Hindman's Theorem, or the Finite Sums Theorem.

Since the set of natural numbers itself is an IP-set and partitions can also be seen as colorings, we can reformulate a special case of Hindman's Theorem in more familiar terms: Suppose the natural numbers are "colored" with n different colors; each natural number gets one and only one of the n colors. Then there exists a color c and an infinite set D of natural numbers, all colored with c, such that every finite sum over D also has color c.

Hindman's Theorem states that the class of IP sets is partition regular.