Asymptotics of Higher-Dimensional General-Position Sets

Definitions and mathematical statement

Let $[n]=\{1,2,\ldots,n\}$ for a positive integer $n$. Under the usual interpretation of the integer grid $[n]^d$ as a subset of ${\mathbb{R}}^d$, for integers $d\geq 1$ and $0\leq k\leq d$, define \[ \alpha_{d,k}(n)=\max\left\{|A|: A\subseteq[n]^d,\quad |A\cap L|\leq k+1\text{ for every affine subspace }L\subseteq{\mathbb{R}}^d\text{ with }\dim L=k\right\}. \] Thus the forbidden configuration consists of $k+2$ distinct points contained in a common $k$-dimensional affine subspace. Configurations whose affine span has dimension less than $k$ are also forbidden, since they are contained in a $k$-dimensional affine subspace.

Problem. For each fixed pair $(d,k)$ in the intended parameter range, determine the asymptotic order of $\alpha_{d,k}(n)$ as $n\to\infty$. In particular, determine the asymptotic order of \[ \alpha_{4,2}(n)=\max\left\{|A|:A\subseteq[n]^4,\quad |A\cap L|\leq 3\text{ for every affine plane }L\subseteq{\mathbb{R}}^4\right\}. \] Determining the asymptotic order means finding a positive function $f_{d,k}(n)$ and constants $c_{d,k},C_{d,k}>0$ such that, for all sufficiently large positive integers $n$, \[ c_{d,k}f_{d,k}(n)\leq\alpha_{d,k}(n)\leq C_{d,k}f_{d,k}(n). \] The parameters $d,k$ are held fixed; the constants and the threshold for $n$ may depend on them.

Unresolved qualifications. The supplied statement does not explicitly specify the admissible integer range for $d,k$ or the ambient scalar field. The definition above makes the conventional real-affine interpretation explicit and is meaningful for $0\leq k\leq d$; whether the source intends the endpoint cases or only a narrower range requires confirmation. The supplied original excerpt also discusses a separate random finite-field problem and contains corrupted displays of background bounds. Those passages do not supply an additional target or a reliably recoverable bound for this statement.

Short English statement

For a fixed dimension $d$ and parameter $k$, find how the largest subset of the integer grid $\{1,\ldots,n\}^d$ with no $k+2$ distinct points in a common affine $k$-dimensional subspace grows as $n\to\infty$. The highlighted case asks for the largest subset of $\{1,\ldots,n\}^4$ with no four points in an affine plane. The real-affine interpretation and the intended parameter range require source confirmation.

Sources

• Local UnsolvedMath registry: supplied source record with permanent ID $30005807$, title "Asymptotics of Higher-Dimensional General-Position Sets," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2024 and names no proposer.
• Supplied report citation: Discrete Geometry (2024), Oberwolfach Reports, DOI 10.4171/owr/2024/3, pp. 137--202. https://doi.org/10.4171/owr/2024/3. The supplied excerpt labels the target "Problem 1."
• Supplied literature source: On higher dimensional point sets in general position. https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/on-higher-dimensional-point-sets-in-general-position/6552A470A725144363409132B022837. No authors or further bibliographic details for this reference are supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that upper and lower bounds are available for $\alpha_{4,2}(n)$ and general fixed $d,k$, but their asymptotic magnitude remains undetermined and newer bounds do not close the gap. This is supplied dated curation derived from the Oberwolfach material, not independently verified here; the record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and says it was reconstructed and checked against the official report. These are supplied verification claims, not verification performed for this draft.
• Background attribution in the supplied excerpt: an earlier upper bound is attributed to Lefmann and an improved bound to Suk--Zeng, reference [15]. A separate random finite-field theorem is attributed to Chen--Liu--Nie--Zeng [4], with partial earlier results attributed to Roche-Newton--Warren [12] and Bhowmick--Roche-Newton [2]. Its container-method background credits the hypergraph container lemma independently to Balogh--Morris--Samotij [1] and Saxton--Thomason [14]. The corresponding full bibliography is not supplied.