Algebra, selections, and additive Ramsey theory

TL;DR

This paper extends classical finite sum theorems to arbitrary topological spaces with Menger's property, using advanced combinatorial and topological methods to find large monochromatic structures and implications for uncountable continuum characteristics.

Contribution

It introduces a new topological and combinatorial framework that generalizes finite sum theorems to uncountable spaces with Menger's property, incorporating Stone--Czech compactification and selection principles.

Findings

Extended Hindman's theorem to Menger spaces

Found large monochromatic structures in open covers

Connected topological properties to uncountable cardinal characteristics

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--Czech 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 $\mathcal{U}$ 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 $V_1 := \bigcup\mathcal{F}_1, V_2 := \bigcup\mathcal{F}_2,\dots$ 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 $F<H$. 2. All vertices $\bigcup_{n\in F}V_n$, for nonempty finite sets $F$, are of the same color. 3. All edges $\bigl\{\,\bigcup_{n\in F}V_n, \bigcup_{n\in H}V_n\,\bigr\}$, for nonempty finite sets $F<H$, have the same color. 4. The family $\{V_1,V_2,\dots\}$ is an open cover of $X$. A self-contained introduction to the necessary parts of the needed theories is provided.