Determine f(4), the maximum size of an acute set in $\mathbb{R}^4$ (and f(5) in $\mathbb{R}^5$)
Statement
A finite set of points in $\mathbb{R}^d$ is an *acute set* if every triple of distinct points spans a triangle all of whose angles are strictly acute (no right or obtuse angles). Let $f(d)$ denote the maximum cardinality of an acute set in $\mathbb{R}^d$. The exact values are known only in dimensions 1, 2, 3: $f(1)=2$, $f(2)=3$, $f(3)=5$. In dimension 4 the current bounds are $9 \le f(4) \le 15$, and in dimension 5 they are $17 \le f(5) \le 31$. Determine $f(4)$ exactly, or improve either bound; likewise for $f(5)$.
Acceptance. ADVANCES (lower bound): exact coordinates (rational or algebraic, with exact-arithmetic verification script) of an acute set of 10 or more points in R^4 (resp. 18+ in R^5) — every triple checked strictly acute; this is a finite, machine-checkable certificate. ADVANCES (upper bound): a proof (human-readable or formal) that f(4) < 15, e.g. f(4) <= 14. FULLY RESOLVES: exact value of f(4) with both a witness and a matching impossibility proof. Numerical near-acute configurations without exact verification do NOT qualify.
Background
Acute sets were introduced by Danzer and Grünbaum (1962), who asked for the maximum size and proved $f(d) \le 2^d - 1$; they conjectured $2d-1$, which stood until Erdős–Füredi's probabilistic exponential construction (1983). The asymptotic question is now settled: Gerencsér and Harangi, 'Acute sets of exponentially optimal size' (arXiv:1705.01171, Discrete Comput. Geom. 2019), construct acute sets of size $2^{d-1}+1$, which is optimal up to the gap with the $2^d-1$ upper bound in fixed small dimensions. The EXACT small-dimensional values have seen no progress since: for $d=4$ the record lower bound is $9 = 2^3+1$ and the upper bound $15 = 2^4-1$. Sources: L. Danzer, B. Grünbaum, 'Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V. L. Klee', Math. Z. 79 (1962) 95–99; V. Harangi, 'Acute sets in Euclidean spaces', SIAM J. Discrete Math. 25 (2011); B. Gerencsér, V. Harangi, arXiv:1705.01171. Problem family appears in Brass–Moser–Pach, 'Research Problems in Discrete Geometry' (Springer 2005), ch. on angle conditions. Vetted open as of 2026-07-06 (no acute-set paper since 2017; bounds current).
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Gerencsér & Harangi, Acute sets of exponentially optimal size (2019) | arxiv |
| REF-02 | Danzer & Grünbaum, Math. Z. 79 (1962) — original problem | doi |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.