  An ordered graph is a graph with a linear ordering on its vertex set. We
prove that for every positive integer $k$, there exists a constant $c_k>0$ such
that any ordered graph $G$ on $n$ vertices with the property that neither $G$
nor its complement contains an induced monotone path of size $k$, has either a
clique or an independent set of size at least $n^{c_k}$. This strengthens a
result of Bousquet, Lagoutte, and Thomass\'e, who proved the analogous result
for unordered graphs.
  A key idea of the above paper was to show that any unordered graph on $n$
vertices that does not contain an induced path of size $k$, and whose maximum
degree is at most $c(k)n$ for some small $c(k)>0$, contains two disjoint linear
size subsets with no edge between them. This approach fails for ordered graphs,
because the analogous statement is false for $k\geq 3$, by a construction of
Fox. We provide further examples how this statement fails for ordered graphs
avoiding other ordered trees as well.
