  In 1933, K. Borsuk proposed the following problem: Can every bounded set in
$\mathbb{E}^n$ be divided into $n+1$ subsets of smaller diameters? In 1965, V.
G. Boltyanski and I. T. Gohberg made the following conjecture: Every bounded
set in an $n$-dimensional metric space can be divided into $2^n$ subsets of
smaller diameters. In this paper, we prove the following result: Every bounded
set in an $n$-dimensional metric space can be divided into $2^{n}((n+1)\log
(n+1)+(n+1)\log \log (n+1)+5n+5)$ subsets of smaller diameters.
