  The probabilistic method is a technique for proving combinatorial existence
results by means of showing that a randomly chosen object has the desired
properties with positive probability. A particularly powerful probabilistic
tool is the Lov\'{a}sz Local Lemma (the LLL for short), which was introduced by
Erd\H{o}s and Lov\'{a}sz in the mid-1970s. Here we develop a version of the LLL
that can be used to prove the existence of continuous colorings. We then give
several applications in Borel and topological dynamics.
  * Seward and Tucker-Drob showed that every free Borel action $\Gamma
\curvearrowright X$ of a countable group $\Gamma$ admits an equivariant Borel
map $\pi \colon X \to Y$ to a free subshift $Y \subset 2^\Gamma$. We give a new
simple proof of this result.
  * We show that for a countable group $\Gamma$, $\mathrm{Free}(2^\Gamma)$ is
weakly contained, in the sense of Elek, in every free continuous action of
$\Gamma$ on a zero-dimensional Polish space. This fact is analogous to the
theorem of Ab\'{e}rt and Weiss for probability measure-preserving actions and
has a number of consequences in continuous combinatorics. In particular, we
deduce that a coloring problem admits a continuous solution on
$\mathrm{Free}(2^\Gamma)$ if and only if it can be solved on finite subgraphs
of the Cayley graph of $\Gamma$ by an efficient deterministic distributed
algorithm (this fact was also proved independently and using different methods
by Greb\'{i}k, Jackson, Rozho\v{n}, Seward, and Vidny\'{a}nszky). This
establishes a formal correspondence between questions that have been studied
independently in continuous combinatorics and in distributed computing.
