Abstract:
In the present paper, a series of problems connecting the Borsuk and Nelson–Erdős–Hadwiger classical problems in combinatorial geometry is considered. The problem has to do with finding the number $\chi(n,a,d)$
equal to the minimal number of colors needed to color an arbitrary set of diameter $d$ in $n$-dimensional Euclidean space in such a way that the distance between points of the same color cannot be equal to $a$. Some new lower bounds for the quantity $\chi(n,a,d)$ are obtained.
Keywords:Borsuk problem, Nelson–Erdős–Hadwiger problem, chromatic number, Stirling formula, infinite graph, Euclidean space, distribution of primes.
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