Clairvoyant embedding in one dimension

Let v,w be in nite 0-1 sequences, and m̃ a positive integer. We say that w is m̃-embeddable in v, if there exists an increasing sequence (ni : i ≥ 0) of integers with n0 = 0, such that 1 ≤ ni − ni−1 ≤ m̃, w(i) = v(ni ) for all i ≥ 1. Let X and Y be coin-tossing sequences. We will show that there is an m̃ with the property that Y is m̃-embeddable into X with positive probability. This answers a question that was open for a while. The proof generalizes somewhat the hierarchical method of an earlier paper of the author on dependent percolation.