We study set systems definable in graphs using variants of logic with different expressive power. Our focus is on the notion of Vapnik-Chervonenkis density: the smallest possible degree of a polynomial bounding the cardinalities of restrictions of such set systems. On one hand, we prove that if $\varphi(\bar x,\bar y)$ is a fixed CMSO$_1$ formula and $\cal C$ is a class of graphs with uniformly bounded cliquewidth, then the set systems defined by $\varphi$ in graphs from $\cal C$ have VC density at most $|\bar y|$, which is the smallest bound that one could expect. We also show an analogous statement for the case when $\varphi(\bar x,\bar y)$ is a CMSO$_2$ formula and $\cal C$ is a class of graphs with uniformly bounded treewidth. We complement these results by showing that if $\cal C$ has unbounded cliquewidth (respectively, treewidth), then, under some mild technical assumptions on $\cal C$, the set systems definable by CMSO$_1$ (respectively, CMSO$_2$) formulas in graphs from $\cal C$ may have unbounded VC dimension, hence also unbounded VC density.

中文翻译：

---

可在树状图中定义的集合系统的 VC 密度

我们使用具有不同表达能力的逻辑变体研究可在图中定义的集合系统。我们的重点是 Vapnik-Chervonenkis 密度的概念：限制此类集合系统的限制基数的多项式的最小可能次数。一方面，我们证明如果 $\varphi(\bar x,\bar y)$ 是一个固定的 CMSO$_1$ 公式，而 $\cal C$ 是一类具有一致有界团宽的图，那么集合系统定义来自 $\cal C$ 的图中 $\varphi$ 的 VC 密度最多为 $|\bar y|$，这是人们可以预期的最小界限。当 $\varphi(\bar x,\bar y)$ 是一个 CMSO$_2$ 公式并且 $\cal C$ 是一类具有一致有界树宽的图时，我们还展示了一个类似的陈述。我们通过证明如果 $\cal C$ 具有无限的团宽（分别为树宽），则对这些结果进行补充，