## Embeddings of ultrametric spaces in finite dimensional structures.(English)Zbl 0639.51018

An ultrametric space (X,d) is a metric space whose distance function d satisfies the strong triangle inequality d(x,z)$$\leq \max (d(x,y),d(y,z))$$. Motivated by applications in theoretical physics and combinatorial optimization the authors study the problem of embedding a finite ultrametric space (X,d) in certain metric spaces such as the space of subsets of an n-elements set, the hypercube of dimension n, $${\mathbb{R}}^ n$$ with the Euclidean distance. Their basic theorem says that in each of these cases $$| X| \leq n+1,$$ this bound being attained. Further, the general embedding problem is examined, and an extension to the case of spaces in which almost every triangle satisfies the strong triangle inequality.
Reviewer: F.D.Veldkamp

### MSC:

 51K05 General theory of distance geometry 05A99 Enumerative combinatorics 82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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### References:

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