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 $2\leq r\leq s$, 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.Comment: 11 pages; the new version includes (as Theorem 1.3) an extension of the main result to more than 2 color

A note on the size Ramsey number of powers of paths

Let $r\geq3$ be an integer such that $r-2$ is a prime power and let $H$ be a connected graph on $n$ vertices with average degree at least $d$ and $\alpha(H)\leq\beta n$, where $0<\beta<1$ is a constant. We prove that the size Ramsey number $$\hat{R}({H};r) > \frac{{nd}}{2}{(r - 2)^2} - C\sqrt n$$ for all sufficiently large $n$, where $C$ is a constant depending only on $r$ and $d$. In particular, for integers $k\ge1$, and $r\ge3$ such that $r-2$ is a prime power, we have that there exists a constant $C$ depending only on $r$ and $d$ such that $\hat{R}(P_{n}^{k}; r)> kn{(r - 2)^2}-C\sqrt n -\frac{{({k^2} + k)}}{2}{(r - 2)^2}$ for all sufficiently large $n$, where $P_{n}^{k}$ is the $kth$ power of $P_n$. We also prove that $\hat{R}(P_n,P_n,P_n)<764.1n$ for sufficiently large $n$. This result improves some results of Dudek and Pra{\l}at (\emph{SIAM J. Discrete Math.}, 31 (2017), 2079--2092 and \emph{Electron. J. Combin.}, 25 (2018), no.3, # P3.35).Comment: 9 pages, 1 figur