[Paper Review] On splitting infinite-fold covers

Márton Elekes, Tamás Mátrai+1 more|ArXiv.org|Nov 14, 2009

Advanced Topology and Set Theory7 references3 citations

TL;DR

This paper investigates the decomposition of infinite-fold covers of sets into disjoint subcovers, focusing on covers by closed sets, convex sets, and polyhedra in topological and ordered spaces. Using set-theoretic methods, including forcing models like the Cohen model and assumptions like Martin’s Axiom, it shows that the possibility of splitting such covers is independent of ZFC, with key results on linearly ordered sets and $[\mathbb{R}^n$ under various geometric constraints.

ABSTRACT

Let $X$ be a set, $\ka$ be a cardinal number and let $\iH$ be a family of subsets of $X$ which covers each $x\in X$ at least $\ka$ times. What assumptions can ensure that $\iH$ can be decomposed into $κ$ many disjoint subcovers? We examine this problem under various assumptions on the set $X$ and on the cover $\iH$: among other situations, we consider covers of topological spaces by closed sets, interval covers of linearly ordered sets and covers of $ eal^{n}$ by polyhedra and by arbitrary convex sets. We focus on these problems mainly for infinite $κ$. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.

Motivation & Objective

• To determine conditions under which an infinite-fold cover of a set can be decomposed into $\kappa$ disjoint subcovers.
• To analyze the splitting problem for covers by closed sets, convex sets, and polyhedra in topological and ordered spaces.
• To investigate the set-theoretic independence of splitting results, particularly in $[\mathbb{R}^n$ and linearly ordered sets.
• To resolve open problems on infinite-fold covers by geometric objects like rectangles, disks, and convex sets.
• To clarify the consistency strength of splitting results using models like the Cohen extension and Martin’s Axiom.

Proposed method

• Uses set-theoretic forcing techniques, including the Cohen model and models satisfying Martin’s Axiom, to construct covers with or without splitting properties.
• Applies elementarity and reflection principles in transitive models to prove the existence of $\kappa$-good colorings for covers.
• Analyzes covers of linearly ordered sets by convex sets, proving maximal decomposability using order-theoretic properties.
• Establishes independence results by showing that in some models (e.g., Cohen extension with GCH), all uncountable-fold closed covers are decomposable, while in others (e.g., under MA), indecomposable covers exist.
• Applies results on interval covers and edge covers of graphs to derive general principles for infinite-fold covers.
• Uses combinatorial coloring techniques, particularly $\kappa$-good colorings, to characterize when a cover can be split into $\kappa$ disjoint subcovers.

Experimental results

Research questions

• RQ1 Under what conditions can an infinite-fold cover of a set by closed sets be decomposed into two disjoint subcovers?
• RQ2 Is the splitting of $\kappa$-fold covers of $\mathbb{R}^n$ by convex sets independent of ZFC?
• RQ3 Can every $\omega$-fold cover of $\mathbb{R}^2$ by translates of a compact convex set be split into two disjoint subcovers?
• RQ4 Is it consistent with ZFC that an $\omega_1$-fold closed cover of $\mathbb{R}$ with $|\mathbf{H}| = \omega_1$ cannot be split into two subcovers?
• RQ5 Does maximal decomposability of covers by convex sets on linearly ordered sets hold in general, and can this be proven more simply than prior results?

Key findings

• For covers of linearly ordered sets by convex sets, maximal decomposability into $\kappa$ disjoint subcovers holds for any infinite $\kappa$, with a simpler proof than prior results.
• The splitting problem for covers of $\mathbb{R}$ by closed sets is independent of ZFC: under Martin’s Axiom, indecomposable $\omega_1$-fold covers exist; in a Cohen extension with GCH, all uncountable-fold closed covers are decomposable.
• The splitting problem for covers of $\mathbb{R}^n$ by convex sets is independent of ZFC, following from the independence result on closed sets.
• In the Cohen model with GCH, every uncountable-fold cover of $\mathbb{R}$ by closed sets is maximally decomposable into $\kappa$ disjoint subcovers.
• For covers of $\mathbb{R}^n$ by convex sets, ZFC does not decide whether $\kappa$-fold covers can always be split, but two ZFC results are established for specific cases.
• The paper shows that for $\kappa$-fold covers of graphs, a complete solution exists for both finite and infinite graphs, generalizing finite combinatorial results.