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arXiv 1606.05628math.COcs.DM
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关于有界度图的有序Ramsey数

On ordered Ramsey numbers of bounded-degree graphs

Martin Balko, Vít Jelínek, Pavel Valtr

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19 pages, 8 figures, minor corrections
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摘要：

有序图是一对$\mathcal{G}=(G,\prec)$，其中$G$是一个图，$\prec$是它的顶点的全序。有序Ramsey数$\overline{R}(\mathcal{G})$是使得有序完全图的边在$N$顶点上的每一个$2$ -染色都包含$\mathcal{G}$的单色副本的最小数$N$.我们证明了对每一个整数$d \geq 3$，几乎每一个$d$ -正则图$G$对$G$的每一个序$\mathcal{G}$都满足$\overline{R}(\mathcal{G}) \geq \frac{n^{3/2-1/d}}{4\log{n}\log{\log{n}}}$.特别地，有3-正则图$G$在$n$顶点上，对于这些图，数字$\overline{R}(\mathcal{G})$在$n$中是超线性的，而不管$G$的顺序$\mathcal{G}$如何。这解决了康伦、福克斯、李和苏达科夫的一个问题。另一方面，我们证明了$G$上的最大度为2的$n$顶点上的每一个图$G$都允许$G$的一个序$\mathcal{G}$，使得$\overline{R}(\mathcal{G})$在$n$中是线性的。我们还证明了几乎每一个具有$n$顶点和区间色数为2的有序匹配$\mathcal{M}$都满足$\overline{R}(\mathcal{M}) \geq cn^2/\log^2{n}$，其中$c$是一个绝对常数。

英文摘要：

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the edges of the ordered complete graph on $N$ vertices contains a monochromatic copy of $\mathcal{G}$. We show that for every integer $d \geq 3$, almost every $d$-regular graph $G$ satisfies $\overline{R}(\mathcal{G}) \geq \frac{n^{3/2-1/d}}{4\log{n}\log{\log{n}}}$ for every ordering $\mathcal{G}$ of $G$. In particular, there are 3-regular graphs $G$ on $n$ vertices for which the numbers $\overline{R}(\mathcal{G})$ are superlinear in $n$, regardless of the ordering $\mathcal{G}$ of $G$. This solves a problem of Conlon, Fox, Lee, and Sudakov. On the other hand, we prove that every graph $G$ on $n$ vertices with maximum degree 2 admits an ordering $\mathcal{G}$ of $G$ such that $\overline{R}(\mathcal{G})$ is linear in $n$. We also show that almost every ordered matching $\mathcal{M}$ with $n$ vertices and with interval chromatic number two satisfies $\overline{R}(\mathcal{M}) \geq cn^2/\log^2{n}$ for some absolute constant $c$.

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