On the number of types in sparse graphs

TLDR: It is proved that for every class of graphs ℒ which is nowhere dense, and for every first order formula φ(x, y), the number of subsets of A|y| which are of the form ū for some valuation ū of x in G is bounded by O(|A||x|ε), which provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes.

Abstract: We prove that for every class of graphs $\mathcal{C}$ which is nowhere dense, as defined by Nesetril and Ossona de Mendez, and for every first order formula $\phi(\bar x,\bar y)$, whenever one draws a graph $G\in \mathcal{C}$ and a subset of its nodes $A$, the number of subsets of $A^{|\bar y|}$ which are of the form $\{\bar v\in A^{|\bar y|}\, \colon\, G\models\phi(\bar u,\bar v)\}$ for some valuation $\bar u$ of $\bar x$ in $G$ is bounded by $\mathcal{O}(|A|^{|\bar x|+\epsilon})$, for every $\epsilon>0$. This provides optimal bounds on the VC-density of first-order definable set systems in nowhere dense graph classes. We also give two new proofs of upper bounds on quantities in nowhere dense classes which are relevant for their logical treatment. Firstly, we provide a new proof of the fact that nowhere dense classes are uniformly quasi-wide, implying explicit, polynomial upper bounds on the functions relating the two notions. Secondly, we give a new combinatorial proof of the result of Adler and Adler stating that every nowhere dense class of graphs is stable. In contrast to the previous proofs of the above results, our proofs are completely finitistic and constructive, and yield explicit and computable upper bounds on quantities related to uniform quasi-wideness (margins) and stability (ladder indices).