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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 $H$, theordered Ramsey number $R_<(H)$ is the minimum $n$ such that everyedge-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 hasorder-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 eachconnected $1$-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 somevertex with order-label $1$.

Upper Bounds for Ordered Ramsey Numbers of Small 1-Orderings