  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 $\xi$, whichwould 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 $p^\prime$ and $p$ respectively, does it
exist $(p', p)$ so that the set of compatible pairs has positive measure? It is
known that this does not happen for $p$ and $p^\prime$ 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 $\mathbb{P}_p {\xi\colon
(\eta_\epsilon,\xi) \text {is compatible}} > 0$ for $p$ small enough, where
$\mathbb{P}_p$ stands for the product Bernoulli measure with parameter $p$.
