  An ordered graph is a graph with a linear ordering on its vertices. The
online Ramsey game for ordered graphs $G$ and $H$ is played on an infinite
sequence of vertices; on each turn, Builder draws an edge between two vertices,
and Painter colors it red or blue. Builder tries to create a red $G$ or a blue
$H$ as quickly as possible, while Painter wants the opposite. The online
ordered Ramsey number $r_o(G,H)$ is the number of turns the game lasts with
optimal play.
  In this paper, we consider the behavior of $r_o(G,P_n)$ for fixed $G$, where
$P_n$ is the monotone ordered path. We prove an $O(n \log_2n)$ bound on
$r_o(G,P_n)$ for all $G$ and an $O(n)$ bound when $G$ is $3$-ichromatic; we
partially classify graphs $G$ with $r_o(G,P_n) = n + O(1)$. Many of these
results extend to $r_o(G,C_n)$, where $C_n$ is an ordered cycle obtained from
$P_n$ by adding one edge.
