Hindman’s Theorem is only a Countable Phenomenon

Abstract

We pursue the idea of generalizing Hindman's Theorem to uncountable cardinalities, 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 generalizations of Hindman's Theorem proposed here tend to fail at all uncountable cardinals. Y ∈F Y F ⊆ X is finite and nonempty of finite unions of elements from X.1991 Mathematics Subject Classification. 03E02 and 05A17 and 05A18 and 20E99.

Algebra, selections, and additive Ramsey theory

“…The following example shows that the Milliken-Taylor Theorem, and thus Hindman's Theorem, is an instance of Theorem 4.6 where Menger's property is trivial: a countable, discrete space. It is illustrative to compare this interpretation with Fernández Bréton's impossibility result [4]: For every set S, there is a 2-coloring of the semigroup Fin(S) of finite subsets of S such that no uncountable subsemigroup of Fin(S) is monochromatic. This demonstrates that the improvement must be on the qualitative side.…”

“…Lemma 3. 4. Let S be a semigroup, and A be a superfilter on S. Every filter generated by a free idempotent chain in A is a free idempotent filter contained in A.…”

Infinite monochromatic sumsets for colourings of the reals

“…2 That is, c(x) only depends on the values of x in its 'support' but not the support itself. 3 Personal communication. 4 I.e., the Generalized Continuum Hypothesis which says that 2 κ = κ + for any infinite cardinal κ.…”

A Note on Hindman-Type Theorems for Uncountable Cardinals

“…The question whether analogs of Hindman's theorem hold for uncountable cardinals received substantial attention in recent years ( [11,13,10,7,8,12,2,6]) 1 . The variants of interest range from the full analog of Hindman's theorem to restrictions thereof where one varies either the number of colours or the set of finite sums that are required to be monochromatic.…”

“…Komjáth [11] and, independently, Soukup and Weiss [13] improved this by ruling out solutions of uncountable cardinality. The second author [7] showed that for any uncountable abelian group G there exists a colouring of G in 2 colours such that no uncountable subset of G has all its finite sums of the same colour. Further negative results on colourings of uncountable abelian groups were proved by the second author and Rinot in a follow-up paper [8], where, e.g., the following theorem is established: Every uncountable abelian group can be coloured with countably many colours, in such a way that every set of finite sums generated by uncountably many elements must be panchromatic (i.e., must contain elements of all possible colours).…”