##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Mikhail Lavrov

#### University of Illinois at Urbana-Champaign

## Ordered size Ramsey number of paths

##### Abstract:

The Erd\H{o}s--Szekeres theorem can be interpreted as saying that in any red-blue edge-coloring of an ordered complete graph on $rs+1$ vertices, there is a red ordered path of length $r$ or a blue ordered path of length $s$. We consider the size Ramsey version of this problem and show that $\tilde{r}(P_r, P_s)$, the least number of edges in an ordered graph with this Ramsey property, satisfies \[ \frac18 r^2 s \le \tilde{r}(P_r, P_s) \le C r^2 s (\log s)^3 \] for any $2 \le r \le s$, where $C>0$ is a constant. This is joint work with J\'ozsef Balogh, Felix Clemen, and Emily Heath.

Host: Jacques Verstraete

### November 26, 2019

### 1:00 PM

### AP&M 7321

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