Borsuk's Problem in Metric Spaces

Abstract: In 1933, K. Borsuk proposed the following problem: Can every bounded set in $\mathbb{E}<sup>n$ 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.