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An ordered graph $G_<$ is a graph with a total ordering $<$ on its vertexset. 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 vertexclasses 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 monotonepath 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 $c_kn/\log n$. Asimilar result holds for ordered graphs containing no induced ordered subgraphisomorphic to a fixed ordered matching.  As a consequence, we give a short combinatorial proof of the followingtheorem of Fox and Pach. There exists a constant $c>0$ such the intersectiongraph $G$ of any collection of $n$ $x$-monotone curves in the plane has abi-clique of size at least $cn/\log n$ or its complement contains a bi-cliqueof size at least $cn$. (A curve is called $x$-monotone if every vertical lineintersects 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 statementdoes not remain true if we replace $\frac14$ by any larger constants.

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