Reconstruction of Point Sets from $k$-Decks

Definitions and mathematical statement

For finite subsets $A,B\subset{\mathbb{R}}^3$, write $A\cong B$ if there exist an orthogonal matrix $U\in O(3)$ and a vector $t\in{\mathbb{R}}^3$ such that $B=\{Ua+t:a\in A\}$. Thus congruence allows reflections as well as rotations and translations. For a positive integer $k$, define the $k$-deck of $A$ to be the multiset \[ D_k(A)=\left\{\!\left\{[S]_{\cong}:S\subseteq A,\ |S|=k\right\}\!\right\}. \] Each unordered $k$-element subset contributes once, and congruent subsets contribute with their multiplicities. The points of $A$ are distinct; no general-position hypothesis is supplied.

The supplied clean statement asks whether a universal positive integer $k$, independent of the finite point set and its cardinality, suffices for reconstruction up to congruence from this deck. There is an unresolved boundary qualification: the source does not specify how sets with fewer than $k$ points are handled, although their $k$-decks are empty. With the additional restriction that the compared sets have at least $k$ points, a precise candidate formulation is \[ \exists k\in{\mathbb{Z}}_{\ge1}\ \forall A,B\subset{\mathbb{R}}^3\text{ finite},\qquad \left(|A|\ge k\ \land\ |B|\ge k\ \land\ D_k(A)=D_k(B)\right) \Longrightarrow A\cong B. \] The cardinality restriction in this display is an explicit possible repair, not a verified source hypothesis. The original opening question instead asks how large $k$ must be for every set of $n$ points to be reconstructible. If cardinality $n$ is supplied separately, its corresponding precise property, for $n\ge k\ge1$, is \[ \forall A,B\subset{\mathbb{R}}^3\text{ finite},\qquad \left(|A|=|B|=n\ \land\ D_k(A)=D_k(B)\right) \Longrightarrow A\cong B. \] The intended boundary convention, whether cardinality is additional reconstruction data, and whether the target is existence of a constant or the optimal value of $k$ require source review.

Short English statement

The question asks whether one fixed number of points per card suffices to reconstruct every finite configuration in ${\mathbb{R}}^3$, up to Euclidean isometry, from the multiplicities of all its congruence classes of cards. The treatment of configurations smaller than a card and the role of a known total cardinality remain unspecified.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30004985$, source code catalogue entry, title "Reconstruction of Point Sets from $k$-Decks." The supplied record lists proposed year 2022 and no named proposer.
• Supplied source citation: "Graph Theory (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/1, pp. 5--78." https://doi.org/10.4171/owr/2022/1. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. The supplied literature list also labels this reference "Oberwolfach Report 1/2022." The differing year labels are retained without reconciliation.
• Supplied literature reference: "Pebody, Radcliffe, and Scott, finite planar subsets." https://digitalcommons.unl.edu/mathfacpub/142/.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and reports that a universal constant deck size is known for planar point sets, whereas in dimensions at least three only logarithmic upper bounds and the general constant-size conjecture are known; it describes the requested sharp $k$ as open. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied record labels its clean statement corrected_verified and says it was checked against the official report and preserves the report's hypotheses, notation, and open target. These are supplied verification claims, not verification performed for this draft.