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On generalized shift graphs

Volume 226 / 2014

Christian Avart, Tomasz Łuczak, Vojtěch Rödl Fundamenta Mathematicae 226 (2014), 173-199 MSC: Primary 05C15; Secondary 05C63. DOI: 10.4064/fm226-2-6

Abstract

In 1968 Erd{ő}s and Hajnal introduced shift graphs as graphs whose vertices are the $k$-element subsets of $[n]=\{1,\dots ,n\}$ (or of an infinite cardinal $\kappa $) and with two $k$-sets $A=\{a_1,\dots ,a_{k}\}$ and $B=\{b_1,\dots ,b_{k}\}$ joined if $a_1< a_2=b_1< a_3=b_2< \cdots < a_k=b_{k-1}< b_k$. They determined the chromatic number of these graphs. In this paper we extend this definition and study the chromatic number of graphs defined similarly for other types of mutual position with respect to the underlying ordering. As a consequence of our result, we show the existence of a graph with interesting disparity of chromatic behavior of finite and infinite subgraphs. For any cardinal $\kappa $ and integer $l$, there exists a graph $G$ with $|V(G)|=\chi (G)=\kappa $ but such that, for any finite subgraph $F\subset G$, $\chi (F)\leq \log_{(l)}|V(F|$, where $\log_{(l)}$ is the $l$-iterated logarithm. This answers a question raised by Erdős, Hajnal and Shelah.

Authors

  • Christian AvartDepartment of Mathematics and Statistics
    Georgia State University
    Atlanta, GA 30303, U.S.A.
    e-mail
  • Tomasz ŁuczakFaculty of Mathematics and CS
    Adam Mickiewicz University
    61-614 Poznań, Poland
    e-mail
  • Vojtěch RödlDepartment of Mathematics and CS
    Emory University
    Atlanta, GA 30322, U.S.A.
    e-mail

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