论文标题
Ramsey的部分顺序图（可比性图）和环理论中的含义
Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
论文作者
论文摘要
对于部分订购的集合$（a，\ le）$，让$ g_a $为简单的，无向的图形，带有顶点套装$ a $，因此，如果$ a \ le a \ le b $或$ b $或$ b \ b \ le a $，则$ in $ in a $中的两个vertices $ a \ neq b \。我们称$ g_a $ the \ emph {部分订单图}或\ emph {可比性图} $ a $。此外，我们说，如果存在部分订购的设置$ a $，则图形$ g $是部分订单图。对于简单，无向图和$ n $，$ m \ ge 1 $的类$ \ MATHCAL {C} $，我们定义Ramsey Numbere $ \ Mathcal {r} _ {\ Mathcal {C}}}（M，N）$与$ \ Mathcal {C cal {C cal {c}由$ r $顶点组成的部分订单图包含一个完整的$ n $ -clique $ k_n $或由$ m $顶点组成的独立集。在本文中，我们根据某些类别的部分订单图确定了Ramsey号码。此外，讨论了拉姆齐数字在环理论中的某些含义。
For a partially ordered set $(A, \le)$, let $G_A$ be the simple, undirected graph with vertex set $A$ such that two vertices $a \neq b\in A$ are adjacent if either $a \le b$ or $b \le a$. We call $G_A$ the \emph{partial order graph} or \emph{comparability graph} of $A$. Further, we say that a graph $G$ is a partial order graph if there exists a partially ordered set $A$ such that $G = G_A$. For a class $\mathcal{C}$ of simple, undirected graphs and $n$, $m \ge 1$, we define the Ramsey number $\mathcal{R}_{\mathcal{C}}(m,n)$ with respect to $\mathcal{C}$ to be the minimal number of vertices $r$ such that every induced subgraph of an arbitrary partial order graph consisting of $r$ vertices contains either a complete $n$-clique $K_n$ or an independent set consisting of $m$ vertices. In this paper, we determine the Ramsey number with respect to some classes of partial order graphs. Furthermore, some implications of Ramsey numbers in ring theory are discussed.
