2020/02/17 by Ayman Badawı, Badawi, Ayman, Roswitha Rissner +1
Mathematics · #05CXX #05D10 #06A06 #13A15 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2002.07134
openalex publication_date 2020/02/17 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
For a partially ordered set (A, \≤), let GA be the simple, undirected\ngraph with vertex set A such that two vertices a \≠ b\∈ A are adjacent\nif either a \≤ b or b \≤ a. We call GA the \partial order graph\nor \comparability graph of A. Further, we say that a graph G is a\npartial order graph if there exists a partially ordered set A such that G =\nGA. For a class \C of simple, undirected graphs and n, m \≥\n1, we define the Ramsey number \R\C(m,n) with respect\nto \C to be the minimal number of vertices r such that every\ninduced subgraph of an arbitrary partial order graph consisting of r vertices\ncontains either a complete n-clique Kn or an independent set consisting of\nm vertices. In this paper, we determine the Ramsey number with respect to\nsome classes of partial order graphs. Furthermore, some implications of Ramsey\nnumbers in ring theory are discussed.\n