Octants are cover-decomposable into many coverings

We prove that octants are cover-decomposable into multiple coverings, i.e., for any k there is an m(k) such that any m(k)-fold covering of any subset of the space with a nite number of translates of a given octant can be decomposed into k coverings. As a corollary, we obtain that any m(k)-fold covering of any subset of the plane with a nite number of homothetic copies of a given triangle can be decomposed into k coverings. Previously only some weaker bounds were known for related problems [21].