Nearly Right-Angled Point Sets

Definitions and mathematical statement

Let \[ \theta_0=\frac{\pi}{2}\left(1-10^{-10000}\right). \] For an integer dimension $n\geq 1$, let $X\subseteq{\mathbb{R}}^n$ be finite. For pairwise distinct points $x,y,z\in X$, define the angle at $y$ by \[ \angle xyz=\arccos\!\left(\frac{\langle x-y,z-y\rangle}{\|x-y\|\,\|z-y\|}\right)\in[0,\pi], \] where the inner product and norm are Euclidean. Assume that \[ \angle xyz\leq\theta_0 \qquad\text{for every ordered triple of pairwise distinct points }x,y,z\in X. \] Thus the restriction applies at every vertex of every triangle determined by three distinct points, and equality with $\theta_0$ is permitted. The proposed assertion is \[ |X|\leq 1.99^n. \] Is this assertion true for every such finite set and every intended dimension?

Unresolved qualification. The source does not explicitly specify the range of dimensions or any sufficiently-large-dimension qualification. The convention $n\geq1$ above makes the ambient dimension explicit, but the intended universal range, including low-dimensional cases and sets for which the triple condition is vacuous, requires review. No asymptotic restriction or exception is being supplied as a correction. This draft follows the supplied reconstructed statement; the raw extraction is incomplete and interleaves neighboring problems.

Short English statement

Must a finite set in Euclidean $n$-space have at most $1.99^n$ points if every angle determined by three distinct points is at most $\frac{\pi}{2}(1-10^{-10000})$? The intended dimension range needs clarification. The supplied record labels the question open; this draft does not independently verify that status.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30005819$, title "Nearly Right-Angled Point Sets," exact source code catalogue entry. The record gives the proposed year as 2024 and supplies no proposer attribution.
• Supplied primary-source citation: Discrete Geometry (2024), Oberwolfach Reports, DOI 10.4171/owr/2024/3, pp. 137--202. https://doi.org/10.4171/owr/2024/3. The record marks its statement corrected_verified and says it was reconstructed and checked against the official report. That verification is supplied provenance, not an independent check performed for this draft.
• Supplied literature source: Kupavskii and Zakharov, The right acute angles problem?. https://arxiv.org/abs/1910.00798.
• Supplied literature source: Oberwolfach Report 3/2024. https://ems.press/content/serial-article-files/48651.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the question open and reports that the right acute angles paper gives bounds near $2^n$ and a lower construction near $(\sqrt{2})^n$, without establishing the proposed $1.99^n$ bound. The background explicitly calls this dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The raw extraction also supplies a neighboring reference: Károlyi and Solymosi, Almost Disjoint Triangles in 3-Space, Discrete & Computational Geometry 28 (2002), pp. 577--583. It appears in the fragment labeled Problem 15 (Eran Nevo), not as a reference or attribution for the angle question. The extraction subsequently begins an incomplete Problem 17 attributed to János Pach.