Metric Transforms and Euclidean Embeddings

It is proved that if <!-- MATH $0 \leqslant c \leqslant 0.72/n$ --> then for any -point metric space , the metric space is isometrically embeddable into a Euclidean space. For -point metric space, <!-- MATH $c = \tfrac{1} {2}{\log _2}\tfrac{3} {2}$ --> is the largest exponent that guarantees the existence of isometric embeddings into a Euclidean space. Such largest exponent is also determined for all -point graphs with "truncated distance".