Sharp Constrained-Drawing Bounds for Closure Operators

Definitions and mathematical statement

A closure operator on ${\mathbb{R}}^2$ is understood in the usual order-theoretic sense as a map $\operatorname{cl}:\mathcal P({\mathbb{R}}^2)\to\mathcal P({\mathbb{R}}^2)$ satisfying, for all $A,B\subseteq{\mathbb{R}}^2$, \[ A\subseteq\operatorname{cl}(A),\qquad A\subseteq B\Longrightarrow\operatorname{cl}(A)\subseteq\operatorname{cl}(B),\qquad \operatorname{cl}(\operatorname{cl}(A))=\operatorname{cl}(A). \] Any further axioms intended for this operator are not specified in the supplied record. Path-connected components are taken in the subspace topology inherited from ${\mathbb{R}}^2$. Assume that \[ \#\{\text{path-connected components of }\operatorname{cl}(S)\}\le b \quad\text{for every finite }S\subseteq{\mathbb{R}}^2. \] Here $b$ is the component-count bound, $K_{1,k}$ is the star with one central vertex and $k$ leaves, and $K_n$ is the complete graph on $n$ vertices. The record does not explicitly specify the admissible integer ranges of $b,k,n$ or its conventions for degenerate cases.

The principal question is whether $kb+1$ points always suffice to obtain a constrained drawing of $K_{1,k}$. If "always suffice" means a guarantee for every point set of that cardinality, the proposed assertion is: for every closure operator satisfying the displayed hypothesis and every finite $P\subseteq{\mathbb{R}}^2$ with $|P|=kb+1$, there exists a constrained drawing of $K_{1,k}$ relative to $P$ and $\operatorname{cl}$. This is only a conditional rendering of the intended quantifiers: the supplied record does not define a constrained drawing or explain how its vertices and edges are constrained by $P$ and $\operatorname{cl}$. Consequently, its existence condition cannot be made into a self-contained mathematical predicate from this record alone.

Two further questions are supplied: what sharper point-count bounds hold for constrained drawings of $K_n$, and whether constrained-drawing arguments extend to settings involving nontrivial homology or homotopy in higher dimensions. The record supplies neither a baseline bound for $K_n$ nor the ambient spaces, dimensions, homological degrees, coefficients, homotopy groups, hypotheses, or target conclusions for the proposed extensions.

The supplied original wording refers to a closure operator on an unspecified space, whereas the clean statement specifies ${\mathbb{R}}^2$. The planar question above follows the clean statement; the scope of any assertion for more general spaces remains unresolved.

Short English statement

The supplied conjecture asks whether $kb+1$ points guarantee a constrained drawing of a star with $k$ leaves when every finite-set closure in ${\mathbb{R}}^2$ has at most $b$ path-connected components. It also asks for better complete-graph bounds and extensions involving higher-dimensional homology or homotopy. The definition of a constrained drawing and the hypotheses for those extensions are missing, so this remains a qualified statement-only draft.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30004513$, source problem code catalogue entry, title "Sharp Constrained-Drawing Bounds for Closure Operators," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2020.
• Supplied report citation: Discrete Geometry (2021), Oberwolfach Reports, pp. 1469--1528, DOI 10.4171/owr/2020/30: https://doi.org/10.4171/owr/2020/30. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: "Oberwolfach Reports 2020/30," https://ems.press/content/serial-article-files/46867.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the available constrained-star bound is quadratic in $b$, while $kb+1$ remains a conjectured improvement. This is supplied dated curation, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The registry assigns the statement status corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and its notation repaired. These are supplied provenance claims; this draft does not independently verify them or claim to have consulted either linked source.