Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory, generalize nowhere dense classes and close them under transductions, i.e. transformations defined by colorings and simple first-order interpretations. In this work we aim to extend some combinatorial and algorithmic properties of nowhere dense classes to monadically stable classes of finite graphs. We prove the following results. - In monadically stable classes the Ramsey numbers $R(s,t)$ are bounded from above by $\mathcal{O}(t^{s-1-\delta})$ for some $\delta>0$, improving the bound $R(s,t)\in \mathcal{O}(t^{s-1}/(\log t)^{s-1})$ known for general graphs and the bounds known for $k$-stable graphs when $s\leq k$. - For every monadically stable class $\mathcal{C}$ and every integer $r$, there exists $\delta > 0$ such that every graph $G \in \mathcal{C}$ that contains an $r$-subdivision of the biclique $K_{t,t}$ as a subgraph also contains $K_{t^\delta,t^\delta}$ as a subgraph. This generalizes earlier results for nowhere dense graph classes. - We obtain a stronger regularity lemma for monadically stable classes of graphs. - Finally, we show that we can compute polynomial kernels for the independent set and dominating set problems in powers of nowhere dense classes. Formerly, only fixed-parameter tractable algorithms were known for these problems on powers of nowhere dense classes.

翻译：粗密的图表没有密集的类别是具有丰富的结构和算法特性的稀有图表, 但是它们无法捕捉到甚至简单的密度图形类。 由模型理论衍生的、 普通化的、 普通化的、 由颜色和简单的第一顺序解释定义的变异性, 也就是说, 我们的目标是将一些低层密集类的组合和算法属性扩展为单调稳定的定量图级。 我们证明了以下结果。 - 在双向稳定的类中, 拉姆赛数字 $R, t 美元来自以上。 由 $\ mathal{ O} (t\ -1\ delta} 开始, 在大约的 美元中, 以 彩色为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为单位, 以美元为美元为单位, 美元为固定的货币为固定。