Multiple Cover Decomposition Thresholds for Planar Sets

Definitions and mathematical statement

For a set $X\subseteq{\mathbb{R}}^2$, an indexed family $\mathcal F=(S_i)_{i\in I}$ of planar sets is an $m$-fold covering of $X$, where $m\in{\mathbb{Z}}_{\geq1}$, if \[ \#\{i\in I:x\in S_i\}\geq m\qquad\text{for every }x\in X. \] A decomposition into $k$ coverings of $X$ is a partition $I=I_1\sqcup\cdots\sqcup I_k$ such that \[ X\subseteq\bigcup_{i\in I_j}S_i\qquad(1\leq j\leq k). \] Indexing permits repeated members to be counted with multiplicity.

The supplied record does not define either cover-decomposable or $m_k(P)$. In particular, it does not specify the allowed geometric copies of $P$, the sets being covered, or the permitted cardinalities of covering families. Consequently, the following is a conditional formalization template, rather than an identification of the source's conventions.

For each $P\subseteq{\mathbb{R}}^2$, let $\mathcal C(P)$ be the intended class of admissible pairs $(X,\mathcal F)$, with $X\subseteq{\mathbb{R}}^2$ and $\mathcal F$ an indexed family of planar sets. For $k\in{\mathbb{Z}}_{\geq2}$, define \[ \begin{split} T_k(P;\mathcal C)=\{m\in{\mathbb{Z}}_{\geq1}:{}&\text{every }(X,\mathcal F)\in\mathcal C(P)\\ &\text{for which }\mathcal F\text{ is an }m\text{-fold covering of }X\\ &\text{admits a decomposition into }k\text{ coverings of }X\}. \end{split} \] Set $\mu_k(P;\mathcal C)=\min T_k(P;\mathcal C)$ when this set is nonempty, and set $\mu_k(P;\mathcal C)=\infty$ otherwise. If the intended definitions identify cover-decomposability with $\mu_2(P;\mathcal C)<\infty$ and $m_k(P)$ with the corresponding finite threshold $\mu_k(P;\mathcal C)$, then the supplied question is: \[ \forall P\subseteq{\mathbb{R}}^2,\qquad \mu_2(P;\mathcal C)<\infty \ \Longrightarrow\ \left(\forall k\in{\mathbb{Z}}_{\geq2},\ \mu_k(P;\mathcal C)<\infty\right). \] The admissibility convention $\mathcal C(P)$ must be recovered from the cited sources before this template can be treated as the precise source statement. No compactness, convexity, measurability, or boundary convention for $P$ is supplied, and none is imposed here.

Short English statement

The question asks whether every cover-decomposable planar set has a finite covering-decomposition threshold $m_k(P)$ for every integer $k\geq2$. The record omits the definitions governing these thresholds, so the displayed formulation is conditional on recovering the intended covering conventions. The supplied curation labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30001887$, source problem code catalogue entry, title Multiple Cover Decomposition Thresholds for Planar Sets. The supplied record lists proposed year 2011 and names no proposer.
• Supplied source citation: Discrete Geometry (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/44, pp. 2459--2548. https://doi.org/10.4171/owr/2011/44. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labelled OWR 44 (2012): https://doi.org/10.4171/owr/2011/44.
• Supplied literature source labelled cover-decomposition survey: https://real.mtak.hu/15290/1/surveyfinal.pdf. No author, title, publication details, or specific passage is supplied for this survey.
• Literature checked at: 2026-08-21. This is supplied dated curation, not an independently verified literature assessment. It labels the problem open and reports that the multiple-covering survey treats finiteness of $m_k(P)$ for every cover-decomposable $P$ as conjectural in this generality. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-status label is corrected_verified; its accompanying explanation says only that TeX notation was enclosed in math delimiters without changing the mathematical content. This does not establish verification of the missing definitions or of mathematical correctness.