  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.
