  Does there exist a Lipschitz injection of $\mathbb{Z}^d$ into the open set of
a site percolation process on $\mathbb{Z}^D$, if the percolation parameter p is
sufficiently close to 1? We prove a negative answer when d=D and also when
$d\geq2$ if the Lipschitz constant M is required to be 1. Earlier work of Dirr,
Dondl, Grimmett, Holroyd and Scheutzow yields a positive answer for d<D and
M=2. As a result, the above question is answered for all d, D and M. Our proof
in the case d=D uses Tucker's lemma from topological combinatorics, together
with the aforementioned result for d<D. One application is an affirmative
answer to a question of Peled concerning embeddings of random patterns in two
and more dimensions.
