Algebra, selections, and additive Ramsey theory

Abstract

Abstract. Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods include, in addition to Hurewicz's game theoretic characterization of Menger's property, extensions of the classic idempotent theory in the Stone-Čech compactification of semigroups, and of the more recent theory of selection principles. This provides strong versions of the mentioned celebrated theorems, where the monochromatic substructures are large, beyond infinitude, in an analytic sense. Reducing the main theorems to the purely combinatorial setting, we obtain nontrivial consequences concerning uncountable cardinal characteristics of the continuum.The main results, modulo technical refinements, are of the following type (definitions provided in the main text): Let X be a Menger space, and U be an infinite open cover of X. Consider the complete graph, whose vertices are the open sets in X. For each finite coloring of the vertices and edges of this graph, there are disjoint finite subsets F 1 , F 2 , . . . of the cover U whose unions V 1 := F 1 , V 2 := F 2 , . . . have the following properties:(1) The sets n∈F V n and n∈H V n are distinct for all nonempty finite sets F < H.(2) All vertices n∈F V n , for nonempty finite sets F , are of the same color.(3) All edges n∈F V n , n∈H V n , for nonempty finite sets F < H, have the same color. (4) The family {V 1 , V 2 , . . . } is an open cover of X. A self-contained introduction to the necessary parts of the needed theories is provided.

Abstract colorings, games and ultrafilters

“…We have n∈N V n = ∅. By the result of Tsaban [35,Lemma 3.4., Theorem 2.7. ], the superfilter A contains an idempotent, large for the sequence V 1 , V 2 , .…”

“…As we already mentioned in the introduction, Tsaban [35] considered theorems about colorings of natural numbers concerning colorings and covers of countable discrete spaces. This approach can be also applied to theorems considered here.…”

“…Tsaban observed that the Hindman Finite Sums Theorem and the Milliken-Tylor Theorem can be viewed as coloring theorems concerning countable covers of countable, discrete space. Every countable space is Menger, and thus these theorems are special instances [35,Example 4.7] of the following result. For a space X, let τ be the topology of X.…”

“…Theorem 1.5 (Tsaban [35,Theorem 4.6]). Let X be a space satisfying the Menger property S fin (O, O) and ∪ be the semigroup operation on τ .…”

Hindman’s Theorem is only a Countable Phenomenon

“…It is worth noting that the generalizations considered in this paper are of a quantitative nature, meaning that they deal only with cardinality, hence the results obtained do not preclude the possibility of obtaining other sorts of generalizations of Hindman's Theorem. For example, Tsaban [13] has shown that Hindman's Theorem may be viewed as a colouring theorem dealing with open covers of a certain countable topological space, and he proved a generalization of this theorem to arbitrary Menger spaces (which can have any arbitrary cardinality, although the objects coloured in this result are countable families of open sets). As another example, Zheng [14] has built on Todorčević's theory of Ramsey Spaces [12] to obtain results where finite subsets of ω together with real numbers are coloured, and monochromatic combinations of FU-sets and perfect subsets of R are obtained.…”