There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of $mathbb{Z}^{d-1}$ joined by a perpendicular copy) into the open set of site percolation on $mathbb{Z}^d$, whenever the parameter p is close enough to 1 or the Lipschitz constant is sufficiently large. This is proved using several new results and techniques involving stochastic domination, in contexts that include a process of independent overlapping intervals on $mathbb{Z}$, and first-passage percolation on general graphs.
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