What is now known as Hindman's Theorem (Theorem 3.1 in [3]) was proved to establish the truth of a conjecture of Graham and Rothschild [2]. It says that if the positive integers are partitioned into finitely many cells, then there is an infinite set of integers all of whose non-empty finite subsets have their sum in the same cell. In [3] van Douwen is credited with realizing that, assuming the Continuum Hypothesis, it is possible to construct an ultrafilter U such that if the positive integers are partitioned into finitely many cells, then there is X ∈ U such that all of the non-empty finite subsets of X have a sum belonging to the same cell. It was noticed by van Douwen that certain ultrafilters had an even stronger property, in that they had a base consisting of all of the finite sums of some set of positive integers. Such ultrafilters are now known as strongly summable ultrafilters. The strongly summable ultrafilters play a significant role in the theory of the semigroup (βN, +). Closely related to Hindman's Theorem is Theorem 3.3 from [3], a result about the semigroup obtained by replacing addition on N with the union operation on the finite subsets of the positive integers. The corresponding ultrafilters, denoted as stable, ordered, union ultrafilters by Blass in [1] are now sometimes called Milliken-Taylor ultrafilters, since Blass showed that they satisfy the Milliken-Taylor Theorem.

Blass also showed that, associated with every stable, ordered, union ultrafilter there is a pair of RK- inequivalent selective ultrafilters and, assuming the Continuum Hypothesis, the correspondence can be reversed. At the end of [1] Blass asks whether his result using the Continuum Hypothesis can be improved by using only the usual axioms of set theory, conjecturing a negative answer. A proof of his conjecture will be presented by constructing a model with two RK-inequivalent selective ultrafilters, but no Milliken-Taylor ultrafilters.

References:

[1] Andreas Blass. Ultrafilters related to Hindman's finite-unions theorem and its extensions. In Logic and combinatorics (Arcata, Calif., 1985), volume 65 of Contemp. Math., pages 89–124. Amer. Math. Soc., Providence, RI, 1987.

[2] R. L. Graham and B. L. Rothschild. Ramsey's theorem for n-parameter sets. Trans. Amer. Math. Soc., 159:257–292, 1971.

[3] Neil Hindman. Finite sums from sequences within cells of a partition of N. J. Combinatorial Theory Ser. A, 17:1–11, 1974.