Let $G$ be a graph on $n$ vertices. Given an integer $t$, we say that a family of sets $\{A_x\}_{x \in V}\subset 2^{[t]}$ is a set representation of $G$ if $$xy \in E(G) \iff A_x \cap A_y = \emptyset$$ Let $\overline{\theta}_1(G)$ be the smallest integer $t$ with such representation. In this talk I will discuss some of the bounds on $\overline{\theta}_1$ for sparse graphs and related problems. Joint work with Basu, Molnar, Rödl and Schacht.