arXiv 2201.076372022-01math.COcs.DM

嵌套匹配的非对角序Ramsey数

On off-diagonal ordered Ramsey numbers of nested matchings

Martin Balko, Marian Poljak

备注：: 17 pages, 7 figures, minor revisions

摘要：

对于两个点集为线性序的图$G^<$和$H^<$，有序Ramsey数$r_<(G^<,H^<)$是最小$N$，使得有序完全图的边在$N$顶点上的每一个红-蓝染色都包含$G^<$的一个红色副本或$H^<$的一个蓝色副本.对于正整数$n$，嵌套匹配$NM^<_n$是$2n$顶点上的有序图，其中每个$i=1,\dots,n$都有边$\{i,2n-i+1\}$。本文改进了Rohatgi得到的有序Ramsey数$r_<(NM^<_n,K^<_3)$的上界，证明了对每一个$n \geq 6$都有$4n+1 \leq r_<(NM^<_n,K^<_3) \leq (3+\sqrt{5})n$，从而否定了Rohatgi的猜想，并确定了$n=4,5$的个数$r_<(NM^<_n,K^<_3)$.作为推论，这给出了对每个$k \geq 3$的$k$ -队列图的最大色数的更强下界.我们还证明了$r_<(NM^<_m,K^<_n)=Θ(mn)$的任意$m$和$n$。我们扩大了经典的拉姆齐善良的概念，有序的情况下，我们试图描述所有连接有序图是$n$ -好的每$n\in\mathbb{N}$。特别是，我们发现了一类新的有序树是$n$ -好的每一个$n \in \mathbb{N}$，扩展所有以前已知的例子。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文