On the chromatic numbers corresponding to exponentially Ramsey sets

In this paper, nontrivial upper bounds on the chromatic numbers of the spaces $\mathbb{R}^n_p=(\mathbb{R}^n, l_p)$ with forbidden monochromatic sets are proved. In the case of forbidden rectangular parallelepiped or a regular simplex, explicit exponential lower bounds on the chromatic numbers are obtained. Exact values of the chromatic numbers of the spaces $\mathbb{R}^n_p$ with forbidden regular simplex in case $p = \infty$ are found.