Stochastic domination and comb percolation

There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of Z(d-1) joined by a perpendicular copy) into the open set of site percolation on 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 Z, and first-passage percolation on general graphs.