On splitting infinite-fold covers

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 $\kappa$ 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 $\real^{n}$ by polyhedra and by arbitrary convex sets. We focus on these problems mainly for infinite $\kappa$. Besides numerous positive and negative results, many questions turn out to be independent of the usual axioms of set theory.

[1]  Jan Mycielski,et al.  Independent sets in topological algebras , 1964 .

[2]  Ram Prakash Gupta,et al.  On decompositions of a multi-graph into spanning subgraphs , 1974 .

[3]  Kenneth Kunen,et al.  Set Theory: An Introduction to Independence Proofs , 2010 .

[4]  K. Falconer.  The geometry of fractal sets , 1985 .

[5]  Arnold W. Miller,et al.  Infinite Combinatorics and Definability , 1989, Ann. Pure Appl. Log..

[6]  Interval covers of a linearly ordered set , 1994 .

[7]  Pertti Mattila,et al.  Geometry of sets and measures in Euclidean spaces , 1995 .

[8]  Haim Judah,et al.  Set Theory: On the Structure of the Real Line , 1995 .

[9]  Kenneth Falconer,et al.  GEOMETRY OF SETS AND MEASURES IN EUCLIDEAN SPACES FRACTALS AND RECTIFIABILITY , 1996 .

[10]  Lajos Soukup,et al.  More Set-theory around the weak Freese-Nation property , 1996 .

[11]  Saharon Shelah,et al.  Sticks and Clubs , 1997, Ann. Pure Appl. Log..

[12]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[13]  William J. Cook,et al.  Combinatorial optimization , 1997 .

[14]  Boaz Tsaban.  The combinatorics of splittability , 2004, Ann. Pure Appl. Log..

[15]  János Pach,et al.  Indecomposable Coverings , 2005, Canadian Mathematical Bulletin.

[16]  Arnold W. Miller,et al.  The number of translates of a closed nowhere dense set required to cover a Polish group , 2006, Ann. Pure Appl. Log..

[17]  Gábor Tardos,et al.  Multiple Coverings of the Plane with Triangles , 2007, Discret. Comput. Geom..

[18]  F. Stephan,et al.  Set theory , 2018, Mathematical Statistics with Applications in R.

[19]  János Pach,et al.  Decomposition of multiple coverings into many parts , 2007, SCG '07.

[20]  Jean Cardinal,et al.  Decomposition of Multiple Coverings into More Parts , 2008, SODA.

[21]  Dömötör Pálvölgyi,et al.  Indecomposable Coverings with Concave Polygons , 2010, Discret. Comput. Geom..

[22]  Géza Tóth,et al.  Convex Polygons are Cover-Decomposable , 2010, Discret. Comput. Geom..