Isometric Embeddings of Finite Metric Spaces / A. I. Oblakova. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2016. № 1. P. 3-9 [Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 1-6].

It is proved that there exists a metric on a Cantor set such that any finite metric space whose diameter does not exceed 1 and the number of points does not exceed n can be isometrically embedded into it. It is also proved that for any m,\,n \in \mathbb N there exists a Cantor set in \mathbb R^m that isometrically contains all finite metric spaces which can be embedded into \mathbb R^m, contain at most n points, and have the diameter at most 1. The latter result is proved for a wide class of metrics on \mathbb R^m and, in particular, for the Euclidean metric.

Key words: metric, isometric embedding, Cantor set.