Algebra, selections, and additive Ramsey theory
Volume 240 / 2018
Abstract
Hindman’s celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, can be viewed as coloring theorems concerning countable covers of countable, discrete sets. These theorems are extended to covers of arbitrary topological spaces with Menger’s classical covering property. The methods include, in addition to Hurewicz’s game-theoretic characterization of Menger’s property, extensions of the classical 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 analytical 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 adjustments, are of the following type (definitions provided in the main text): Let $X$ be a Menger space, and $\mathcal{U}$ 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 $\mathcal{F}_1,\mathcal{F}_2,\dots$ of the cover $\mathcal{U}$ whose unions $V_1 := \bigcup\mathcal{F}_1$, $V_2 := \bigcup\mathcal{F}_2,\ldots$ have the following properties:
(1) The sets $\bigcup_{n\in F}V_n$ and $\bigcup_{n\in H}V_n$ are distinct for all nonempty finite sets $F \lt H$.
(2) All vertices $\bigcup_{n\in F}V_n$ for nonempty finite sets $F$ have the same color.
(3) All edges $\{\bigcup_{n\in F}V_n, \bigcup_{n\in H}V_n\}$ for nonempty finite sets $F \lt H$ have the same color.
(4) The family $\{V_1,V_2,\dots\}$ forms a cover of $X$.
A self-contained introduction to the necessary parts of the needed theories is included.
