Abstract
For two ordered graphs $G^<$ and $H^<$, the \emph{ordered Ramsey number} $r_<(G^<,H^<)$ is the minimum $N$ such that every red-blue coloring of the edges of the ordered complete graph $K^<_N$ contains a red copy of~$G^<$ or a blue copy of $H^<$. For $n \in \mathbb{N}$, a \emph{nested matching} $NM^<_n$ is the ordered graph on $2n$ vertices with edges $\{i,2n-i+1\}$ for every $i=1,\dots,n$.
We improve bounds on the numbers $r_<(NM^<_n,K^<_3)$ obtained by Rohatgi, we disprove his conjecture about these numbers, and we determine them exactly for $n=4,5$. This gives a stronger lower bound on the maximum chromatic number of $k$-queue graphs for every $k \geq 3$.
We expand the classical notion of Ramsey goodness to the ordered case and we attempt to characterize all connected ordered graphs that are $n$-good for every $n\in\mathbb{N}$. In particular, we discover a new class of such ordered trees, extending all previously known examples.