Abstract:
Two classical problems of combinatorial geometry, the Borsuk problem about splitting sets into parts of smaller diameter and the Erdös–Hadwiger problem about coloring Euclidean space, are studied. New asymptotic estimates are obtained for the quantities $f(d)$ (the minimal number of parts of smaller diameter into which any bounded set in $\mathbb R^d$ can be decomposed) and $\chi(\mathbb R^d)$ (the minimal number of colors required to color all points $\mathbb R^d$ so that any points at distance 1 from each other have different colors), which are the main objects of study in these problems.
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