Transfinite approximation of Hindman's theorem

Abstract

Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the finite analogue stating that for any n, r, there is a k such that for any r-coloring of [1, k], there is a set of n integers all of whose finite sums belong to the same color. We extend the finite form of Hindman's Theorem to an α-Hindman Theorem for each countable ordinal α. These α-statements approximate Hindman's Theorem in the sense that the full fledged theorem is equivalent to the transfinite version holding for every countable ordinal. We then give a proof of Hindman's Theorem by directly proving these transfinite approximations. © 2011 Hebrew University Magnes Press.