On off-diagonal ordered Ramsey numbers of nested matchings

TL;DR

This paper advances understanding of ordered Ramsey numbers for nested matchings and complete graphs, disproves a conjecture, and explores ordered Ramsey goodness, revealing new classes of ordered trees with universal properties.

Contribution

It improves bounds on ordered Ramsey numbers for nested matchings and triangles, disproves Rohatgi's conjecture, and characterizes ordered trees that are n-good for all n.

Findings

Bounds on r_<(NM^<_n,K^<_3) are tightened, disproving Rohatgi's conjecture.

Exact values of r_<(NM^<_n,K^<_3) are determined for n=4,5.

New classes of ordered trees are identified as n-good for all n.

Abstract

For two graphs $G^<$ and $H^<$ with linearly ordered vertex sets, the ordered Ramsey number $r_<(G^<,H^<)$ is the minimum $N$ such that every red-blue coloring of the edges of the ordered complete graph on $N$ vertices contains a red copy of $G^<$ or a blue copy of $H^<$. For a positive integer $n$, a 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 ordered Ramsey numbers $r_<(NM^<_n,K^<_3)$ obtained by Rohatgi, we disprove his conjecture by showing $4n+1 \leq r_<(NM^<_n,K^<_3) \leq (3+\sqrt{5})n$ for every $n \geq 6$, and we determine the numbers $r_<(NM^<_n,K^<_3)$ exactly for $n=4,5$. As a corollary, this gives stronger lower bounds on the maximum chromatic number of $k$-queue graphs for every $k \geq 3$. We also prove $r_<(NM^<_m,K^<_n)=\Theta(mn)$ for arbitrary $m$ and $n$. 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 ordered trees that are $n$-good for every $n \in \mathbb{N}$, extending all the previously known examples.