The maximum number of unit distances among \(n\) points in dimension four
This paper completes the determination of the exact maximum number of unit distances among \(n\) points in euclidean fourdimensional space. This problem, which is a very difficult open problem in dimensions two and three, becomes much simpler in higher dimensions and was asymptotically solved long ago by \textit{P. Erdős} [Can. J. Math. 19, 968-971 (1967; Zbl 0161.20603)] and \textit{P. Erdős} and \textit{J. Pach} [Combinatorica 10, No. 3, 261-269 (1990; Zbl 0722.52009)]. In [\textit{P. Braß}, Bolyai Soc. Math. Stud. 6, 277-290 (1997; Zbl 0885.52015)] the reviewer showed that it is even possible to solve the problem exactly, for dimension four, where he obtained the structure of the extremal sets and their edge-number, depending on the existence of a solution of a trigonometric diophantine equation (see the last cited paper for a correction of one case). This diophantine equation is solved in this paper by van Wamelen, so that the complete answer is: the maximum number of unit distances among \(n\) points in dimension four is \(\lfloor{1\over 4}n^2\rfloor +n\) if \(n\) is divisible by 8 or 10, and one less for all other \(n\).
scientific article; zbMATH DE number 1460291