Radon Numbers for Unions of Convex Sets

Definitions and mathematical statement

For the following working formulation, take $d,s,t\in{\mathbb{Z}}_{\geq 1}$. The supplied record describes $r$-convex sets as unions of a bounded number of convex sets but does not give a formal definition. Interpret this as the family \[ \mathcal U_r(d)=\left\{\bigcup_{i=1}^{m}K_i:0\leq m\leq r,\ K_i\subseteq{\mathbb{R}}^d\text{ is convex for every }i\right\}, \] where the union with $m=0$ is empty. Convexity means that $(1-\lambda)x+\lambda y\in K_i$ whenever $x,y\in K_i$ and $\lambda\in[0,1]$. No closedness, boundedness, or connectedness of the union is required.

For $n\in{\mathbb{Z}}_{\geq1}$, let $\mathcal R_{d,s,t}(n)$ denote the property \[ \begin{aligned} &\forall P\subseteq{\mathbb{R}}^d\text{ with }|P|=n,\ \exists A,B\subseteq P:\\ &\quad P=A\sqcup B,\quad A\ne\varnothing,\quad B\ne\varnothing,\\ &\quad \forall S\in\mathcal U_s(d)\ \forall T\in\mathcal U_t(d),\quad (A\subseteq S\ \land\ B\subseteq T)\Longrightarrow S\cap T\ne\varnothing. \end{aligned} \] Thus the partition is chosen before the universally quantified containing sets. There is no general-position hypothesis on $P$. Under the convention that the empty set belongs to each $\mathcal U_r(d)$, the intersection requirement itself excludes empty parts.

Determine the least integer \[ f(d,s,t)=\min\{n\in{\mathbb{Z}}_{\geq1}:\mathcal R_{d,s,t}(n)\}, \] whenever this set is nonempty. The accompanying polynomial-growth question is interpreted as asking whether there exist constants $C>0$ and $k\in{\mathbb{Z}}_{\geq0}$, independent of $d,s,t$, such that this minimum exists and \[ f(d,s,t)\leq C(d+s+t)^k \qquad\text{for every }d,s,t\in{\mathbb{Z}}_{\geq1}. \] The supplied Radon-theorem comparison gives $f(d,1,1)=d+2$, with sharpness witnessed by the vertices of a $d$-dimensional simplex.

The parameter ranges, the precise meaning of $r$-convexity, and the interpretation of polynomial growth above are explicit conventions requiring review against the complete mathematical source. The supplied dated curation labels the problem partially solved and attributes a near-optimal polynomial upper bound to Alon and Smorodinsky, while reporting that the exact least value remains unknown. These are supplied literature assessments, not independently verified conclusions of this draft.

Short English statement

Find the smallest number of points in ${\mathbb{R}}^d$ that always permits a partition into two nonempty parts such that every union of at most $s$ convex sets containing the first part intersects every union of at most $t$ convex sets containing the second. The source also asks for polynomial growth in the parameters. Its supplied dated assessment reports a polynomial upper bound but leaves the exact minimum unresolved; the conventions and literature claims require review.

Sources

• Local UnsolvedMath registry, supplied source record, permanent ID $30006538$, title Radon Numbers for Unions of Convex Sets, exact source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and carries the supplied labels partially_solved and corrected_verified.
• Combinatorics (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/1, pp. 5--79. Supplied report reference and literature-source link: https://doi.org/10.4171/owr/2026/1.
• Alon--Smorodinsky, Extended VC-dimension and Radon and Tverberg type theorems. Supplied literature-source link: https://arxiv.org/abs/2506.17777.
• The supplied original excerpt attributes the motivating problem to Kalai in the 1970s and says that later surveys reiterated it. Its references labeled [1] and [2] are not accompanied by a bibliography in the supplied excerpt; their exact identification is unresolved.
• Literature checked at: 2026-08-22. This exact date records supplied dated curation, not independent verification. That curation credits Alon and Smorodinsky with the first near-optimal polynomial upper bound, describes the polynomial-growth question as resolved, and reports that the exact least value remains unknown.
• The registry's supplied statement-verification note says that the official Oberwolfach report was checked and the question isolated and its notation repaired. This is a source-verification claim supplied by the registry; this draft does not independently verify it.