Abstract For a partially ordered set ( A , ≤ ) (A,\le ) , let G A {G}_{A} be the simple, undirected graph with vertex set A such that two vertices a ≠ b ∈ A a\ne b\in A are adjacent if either a ≤ b a\le b or b ≤ a b\le a . We call G A {G}_{A} the partial order graph or comparability graph of A. Furthermore, we say that a graph G is a partial order graph if there exists a partially ordered set A such that G = G A G={G}_{A} . For a class C {\mathcal{C}} of simple, undirected graphs and n, m ≥ 1 m\ge 1 , we define the Ramsey number ℛ C ( n , m ) { {\mathcal R} }_{{\mathcal{C}}}(n,m) with respect to C {\mathcal{C}} to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in C {\mathcal{C}} consisting of r vertices contains either a complete n-clique K n {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.

中文翻译：

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偏序图（可比性图）的拉姆齐数和环论中的含义

摘要 对于偏序集 ( A , ≤ ) (A,\le ) ，令 GA {G}_{A} 是顶点集 A 的简单无向图，使得两个顶点 a ≠ b ∈ A a\ne b \in A 相邻，如果 a ≤ ba\le b 或 b ≤ ab\le a 。我们称 GA {G}_{A} 为 A 的偏序图或可比性图。 此外，如果存在偏序集 A 使得 G = GAG={G}，我们称图 G 是偏序图_{一种} 。对于 C 类 {\mathcal{C}} 的简单无向图和 n, m ≥ 1 m\ge 1 ，我们定义 Ramsey 数 ℛ C ( n , m ) { {\mathcal R} }_{{\ mathcal{C}}}(n,m) 相对于 C {\mathcal{C}} 是顶点数 r 的最小数量，使得 C {\mathcal{C}} 中任意图的每个诱导子图由r 个顶点包含一个完整的 n-clique K n {K}_{n} 或一个由 m 个顶点组成的独立集合。在本文中，我们确定了关于某些偏序图类的 Ramsey 数。此外，讨论了环理论中拉姆齐数的一些含义。