Transfinite Approximation of Hindman's Theorem

TL;DR

This paper extends Hindman's Theorem to a transfinite setting for all countable ordinals, establishing an equivalence and providing a new proof via transfinite approximations.

Contribution

It introduces a transfinite extension of Hindman's Theorem for each countable ordinal and proves their equivalence, offering a novel proof approach.

Findings

Hindman's Theorem is equivalent to its transfinite approximations for all countable ordinals.

The paper provides a new proof of Hindman's Theorem using transfinite methods.

Establishes a framework connecting finite, infinite, and transfinite combinatorial properties.

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 corresponding finite form, stating that in any finite coloring of the integers there are arbitrarily long finite sets with the same property. We extend the finite form of Hindman's Theorem to a "transfinite" version for each countable ordinal, and show that Hindman's Theorem is equivalent to the appropriate transfinite approximation holding for every countable ordinal. We then give a proof of Hindman's Theorem by directly proving these transfinite approximations.