A partition property of simplices in Euclidean space

Frankl, P.; Rödl, V.

doi:10.1090/S0894-0347-1990-1020148-2

A partition property of simplices in Euclidean space
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by P. Frankl and V. Rödl

J. Amer. Math. Soc. 3 (1990), 1-7

DOI: https://doi.org/10.1090/S0894-0347-1990-1020148-2

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Abstract:

Given the vertex set $A$ of a nondegenerate simplex in ${R^d}$, it is shown that for some positive $\varepsilon = \varepsilon (A)$ and every partition of ${R^n}$ into fewer than ${(1 + \varepsilon )^n}$ parts, one of the parts must contain a set congruent to $A$. This solves a fifteen-year-old problem of Erdös et al. [E].

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