Acute sets (Q1739206)

An acute set referred to in the title is a set of points in the $d$ -dimensional Euclidean space $\mathbb{R}^d$ such that any three points in the set form an acute triangle. \par The problem of \textit{L. Danzer} and \textit{B. Grünbaum} [Math. Z. 79, 95--99 (1962; Zbl 0188.27602)] is to determine the size of a maximal acute set of points $f(d)$. Their conjecture from 1962 that $f(d)\geq 2d-1$ was shown to be false in high dimensions by \textit{P. Erdős} and \textit{Z. Füredi} [Ann. Discrete Math. 17, 275--283 (1983; Zbl 0534.52007)]. \par The author of this paper proves that $f(d+2)\geq2f(d)$, and then he proves inductively that $f(d)\geq F_{d+1}$, where $F_d$ is the $d$-th Fibonacci number. Both proofs are constructive and explicit.