I have seen several references to the so-called ** Extension Theorem** in the context of tilings of Euclidean space.
E.g. in "The Local Theorem for Monotypic Tilings" one reads

The Extension Theorem [...] gives a criterion for a finite monohedral complex of polytopes to be extendable to a global isohedral tiling of space.

I have a hard time tracking down the exact statement of this theorem. I found some sources (see below), but these are available only in Russian (despite the English titles).

- N. Dolbilin, "The Extension Theorem".
- N.P. Dolbilin and V.S. Makarov, "The extension theorem in the theory of regular tilings and its applications".