# Upper Bounds for Ordered Ramsey Numbers of Small 1-Orderings

## TL;DR

This paper systematically studies ordered Ramsey numbers for small 1-orderings, providing upper bounds for various small graphs and paths, and extending results to generalized ordered Ramsey numbers involving vertices with specific order-labels.

## Contribution

It offers the first comprehensive bounds for 1-orderings of small graphs and extends these results to paths and generalized cases, advancing understanding of ordered Ramsey theory.

## Key findings

- Upper bounds for ordered Ramsey numbers of connected 1-orderings on 4 vertices.
- Linear bound for ordered Ramsey numbers of 1-orderings of paths.
- Upper bounds for generalized ordered Ramsey numbers involving vertices with order-label 1.

## Abstract

A $k$-ordering of a graph $G$ assigns distinct order-labels from the set $\{1,\ldots,|G|\}$ to $k$ vertices in $G$. Given a $k$-ordering $H$, the ordered Ramsey number $R_<(H)$ is the minimum $n$ such that every edge-2-coloring of the complete graph on the vertex set $\{1, \ldots, n\}$ contains a copy of $H$, the $i$th smallest vertex of which either has order-label $i$ in $H$ or no order-label in $H$. This paper conducts the first systematic study of ordered Ramsey numbers for $1$-orderings of small graphs. We provide upper bounds for $R_<(H)$ for each connected $1$-ordering $H$ on $4$ vertices. Additionally, for every $1$-ordering $H$ of the $n$-vertex path $P_n$, we prove that $R_<(H) \in O(n)$. Finally, we provide an upper bound for the generalized ordered Ramsey number $R_<(K_n, H)$ which can be applied to any $k$-ordering $H$ containing some vertex with order-label $1$.