# Ordered size Ramsey number of paths

@article{Balogh2020OrderedSR, title={Ordered size Ramsey number of paths}, author={J. Balogh and Felix Christian Clemen and Emily Heath and M. Lavrov}, journal={Discret. Appl. Math.}, year={2020}, volume={276}, pages={13-18} }

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… Expand

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On edge-ordered Ramsey numbers

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It is proved that for every edge-ordered graph $H$ on $n$ vertices, the authors have $r_{edge}(H;q) \leq 2^{c^qn^{2q-2}\log^q n}$, where $c$ is an absolute constant. Expand

A Strengthening of the Erd\H{o}s-Szekeres Theorem

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The Erdős-Szekeres Theorem stated in terms of graphs says that any red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$ contains a red copy of the monotone increasing path with $r$… Expand

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