On set systems definable in sparse graph classes, discrepancy, and quantifier elimination

We study the (hereditary) discrepancy of definable set systems, which is a natural measure for their inherent complexity and approximability. We establish a strong connection between the hereditary discrepancy and quantifier elimination over signatures with only unary relation and function symbols. We prove that set systems definable in theories (over such signatures) with quantifier elimination have constant hereditary discrepancy. We derive that set systems definable in bounded expansion classes, which are very general classes of uniformly sparse graphs, have bounded hereditary discrepancy. We also derive that nowhere dense classes, which are more general than bounded expansion classes, in general do not admit quantifier elimination over a signature extended by an arbitrary number of unary function symbols. We show that the set systems on a ground set $U$ definable on a monotone nowhere dense class of graphs $\mathscr C$ have hereditary discrepancy at most $|U|^{c}$ (for some~$c<1/2$) and that, on the contrary, for every monotone somewhere dense class $\mathscr C$ and for every positive integer $d$ there is a set system of $d$-tuples definable in $\mathscr C$ with discrepancy $\Omega(|U|^{1/2})$. We further prove that if $\mathscr C$ is a class of graphs with bounded expansion and $\phi(\bar x;\bar y)$ is a first-order formula, then we can compute in polynomial time, for an input graph $G\in\mathscr C$, a mapping $\chi:V(G)^{|\bar x|}\rightarrow\{-1,1\}$ witnessing the boundedness of the discrepancy of the set-system defined by~$\phi$, an $\varepsilon$-net of size $\mathcal{O}(1/\varepsilon)$, and an $\varepsilon$-approximation of size $\mathcal{O}(1/\varepsilon)$.