{
  "id": "2002.07134",
  "version": "v1",
  "published": "2020-02-17T18:54:05.000Z",
  "updated": "2020-02-17T18:54:05.000Z",
  "title": "Ramsey numbers of partial order graphs and implications in ring theory",
  "authors": [
    "Ayman Badawi",
    "Roswitha Rissner"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "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} 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-17T18:54:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "13A15",
      "06A06",
      "05D10"
    ],
    "keywords": [
      "ramsey number",
      "ring theory",
      "implications",
      "partially ordered set",
      "arbitrary partial order graph consisting"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}