Ordered Size Ramsey Number of Paths

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 $2\le r\le s$, 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.