On generalized shift graphs
Tom 226 / 2014
Streszczenie
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.
