  An ordered graph is a simple graph with an ordering on its vertices. Define
the ordered path $P_n$ to be the monotone increasing path with $n$ edges. The
ordered size Ramsey number $\tilde{r}(P_r,P_s)$ is the minimum number $m$ for
which there exists an ordered graph $H$ with $m$ edges such that every
two-coloring of the edges of $H$ contains a red copy of $P_r$ or a blue copy of
$P_s$. For $2\leq r\leq s$, we show $\frac{1}{8}r^2s\leq \tilde{r}(P_r,P_s)\leq
Cr^2s(\log s)^3$, where $C>0$ is an absolute constant. This problem is
motivated by the recent results of Buci\'c-Letzter-Sudakov and Letzter-Sudakov
for oriented graphs.
