In this talk we will investigate the interactions between combinatorial and dynamical properties of actions of countable groups. The starting point of our investigation is the result of Seward and Tucker-Drob which says that if $\Gamma$ be a countably infinite group, then every free Borel action of $\Gamma$ on a Polish space $X$ admits a Borel equivariant map $\pi$ to the free part of the Bernoulli shift $2^\Gamma$. Our goal is to generalize this result by putting additional combinatorial restrictions on the image of $\pi$. For instance, can we ensure that $\pi(x)$ is a proper coloring of the Cayley graph of $\Gamma$ for all $x \in X$? More generally, can we guarantee that the image of $\pi$ is contained in a given subshift of finite type? The main result of this talk is a positive answer to this question in a very general setting. The main tool used in the proof of our result in a probabilistic technique for constructing continuous functions with desirable properties, namely a continuous version of the Lov\'{a}sz Local Lemma.