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".



What does the extension theorem for tilings state?
The first source is in English in Discrete Mathematics. You can find it here


* Nikolai P. Dolbilin, *The Extension Theorem*, Discrete Mathematics **221** Issues 1–3 (2000) pp 43–59, <https://doi.org/10.1016/S0012-365X(99)00385-4>.


