On the compatibility of binary sequences

An ordered pair of semi-infinite binary sequences (eta, xi) is said to be compatible if there is a way of removing a certain number (possibly infinite) of ones from eta and zeroes from xi that would map both sequences to the same semi-infinite sequence. This notion was introduced by Peter Winkler, who also posed the following question: eta and xi being independent i.i.d. Bernoulli sequences with parameters p ' and p, respectively, does there exist (p ', p) so that the set of compatible pairs has positive measure? It is known that this does not happen for p and p ' very close to 1/2. In the positive direction, we construct, for any epsilon > 0, a deterministic binary sequence eta(epsilon) whose set of zeroes has Hausdorff dimension larger than 1 - epsilon and such that P-p{xi : (eta(epsilon), xi) is compatible } > 0 for p small enough, where P-p stands for the product Bernoulli measure with parameter p. (c) 2014 Wiley Periodicals, Inc.