No exceptional words for Bernoulli percolation

Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as percolation of words. We give a positive answer to their Open Problem 2: almost surely, all words are seen for site percolation on Z(3) with parameter p = 1/2. We also extend this result in various directions, proving the same result on Z(d) , d >= 3, for any value p is an element of (p(c)(site) (Z(d)), 1 - p(c)(site) (Z(d))), and for restrictions to slabs. Finally, we provide an explicit estimate on the probability to find all words starting from a finite box.