Order (2018) 35:83–91
Hindman’s Theorem is only a Countable Phenomenon
David J. Fern
Received: 13 January 2016 / Accepted: 24 November 2016 / Published online: 6 December 2016
© Springer Science+Business Media Dordrecht 2016
Abstract We pursue the idea of generalizing Hindman’s Theorem to uncountable cardinal-
ities, by analogy with the way in which Ramsey’s Theorem can be generalized to weakly
compact cardinals. But unlike Ramsey’s Theorem, the outcome of this paper is that the natural gen-
eralizations of Hindman’s Theorem proposed here tend to fail at all uncountable cardinals.
Keywords Ramsey-type theorem · Hindman’s theorem · Uncountable cardinals ·
-system lemma · Semigroups · Abelian groups
Hindman’s theorem is one of the most famous and interesting examples of a so-called
Ramsey-type theorem, a theorem about partitions.
Theorem 1 (Hindman ) For every partition N = A
of the set of natural numbers
into two cells, there exists an infinite X ⊆ N such that for some i ∈ 2, FS(X) ⊆ A
FS(X) denotes the set
F ⊆ X is finite and nonempty
of all finite sums of elements of X).
The author was partially supported by postdoctoral fellowship number 263820 from the Consejo
Nacional de Ciencia y Tecnolog
ıa (Conacyt), Mexico.
David J. Fern
Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street,
Ann Arbor, MI 481091043, USA