On off-diagonal ordered Ramsey numbers of nested matchings

TL;DR

The paper studies off-diagonal ordered Ramsey numbers for nested matchings against complete graphs and extends Ramsey goodness to the ordered setting. It tightens the bound for $r_<(NM^<_n,K^<_3)$ to $\le (3+\sqrt{5})n+1$ and proves a universal lower bound $4n+1$ for $n\ge 6$, with exact values for $n=4,5$, and establishes $r_<(NM^<_m,K^<_n)=\Theta(mn)$. It introduces and analyzes the notion of good ordered graphs, identifying monotone caterpillar graphs as a broad, conjecturally complete class via join operations and a forbidden-subgraph characterization, supported by SAT-based computations for small cases. These results connect to queue-number related chromatic properties of $k$-queue graphs and provide a framework for further understanding ordered Ramsey goodness and its structural implications.

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 $r_<(NM^<_m,K^<_n)=Θ(mn)$ 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.

Figures (7)

• Figure 1: (a) The ordered star graph $S_{4,3}^<$. (b) An example of a monotone caterpillar graph $S^<_{1,3}+S^<_{3,1}+S^<_{1,2}+S^<_{1,4}$. Note that this ordered graph can be also written as, for example, $S^<_{1,3} + S^<_{3,2}+S^<_{1,4}$.
• Figure 2: Any ordered graph that does not contain any of these four ordered graphs as an ordered subgraph is a monotone caterpillar graph.
• Figure 3: Forbidden connected ordered subgraphs for monotone caterpillar graphs. Also, these are all ordered trees on four vertices that are not monotone caterpillar graphs.
• Figure 4: (a) An example of an anti-diagonal in the matrix $A$ for $N=9$. (b) A construction of two disjoint routes in $A$ for $N=9$ and $n=3$, where red entries are 1-entries and blue entries are 0-entries. Each anti-diagonal achieves the maximum possible number of 1-entries, which leads to an ordered graph that does not contain a copy of $NM_3^<$ and has the maximum number of edges. The dashed paths denote the routes without their endpoints on the diagonal.
• Figure 5: The matrix representation of the coloring $\chi$ of the edges of $K^<_{4n}$ for $n=6$.

Theorems & Definitions (21)

• Theorem 1: balkoconlon
• Conjecture 4: rohatgi
• Theorem 6
• Theorem 7
• Corollary 8
• Theorem 9
• Theorem 10
• Proposition 11
• Theorem 12
• Corollary 13