Ordered graphs and large bi-cliques in intersection graphs of curves

An ordered graph $G_<$ is a graph with a total ordering $<$ on its vertex set. A monotone path of length $k$ is a sequence of vertices $v_1<v_2<\ldots<v_k$ such that $v_iv_{j}$ is an edge of $G_<$ if and only if $|j-i|=1$. A bi-clique of size $m$ is a complete bipartite graph whose vertex classes are of size $m$. We prove that for every positive integer $k$, there exists a constant $c_k>0$ such that every ordered graph on $n$ vertices that does not contain a monotone path of length $k$ as an induced subgraph has a vertex of degree at least $c_kn$, or its complement has a bi-clique of size at least $c_kn/\log n$. A similar result holds for ordered graphs containing no induced ordered subgraph isomorphic to a fixed ordered matching. As a consequence, we give a short combinatorial proof of the following theorem of Fox and Pach. There exists a constant $c>0$ such the intersection graph $G$ of any collection of $n$ $x$-monotone curves in the plane has a bi-clique of size at least $cn/\log n$ or its complement contains a bi-clique of size at least $cn$. (A curve is called $x$-monotone if every vertical line intersects it in at most one point.) We also prove that if $G$ has at most $\left(\frac14 -\epsilon\right){n\choose 2}$ edges for some $\epsilon>0$, then $\overline{G}$ contains a linear sized bi-clique. We show that this statement does not remain true if we replace $\frac14$ by any larger constants.