The Generalised Shift Graph

In 1968, Erd"os defined the Shift Graph as the graph whose vertices are the $k$-element subsets of $[n]=\{0,1,2,\mathrm{.}\mathrm{.}\mathrm{.},n-1\}$ such that $A=\{a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_k\}$ and $B=\{b_1,\mathrm{.}\mathrm{.}\mathrm{.},b_k\}$ are neighbours iff $a_1<b_1=a_2<b_2=a_3<… <b_{n-1}=a_n<b_n$. In the paper $\mathit{\text{On}}\mathit{\text{the}}\mathit{\text{Generalised}}\mathit{\text{Shift}}\mathit{\text{Graph}}$, Avart, Luczac and R"odl extend this definition to include all possible arrangements of the $a_{i}s$ and $b_{i}s$, known as $\mathit{\text{types}}$. In this paper, we will consider a selection of these types and study the corresponding graphs. We are interested in to what extent the graphs $G(S,\tau )$ and $G(S^{\mathrm{\prime }},\tau )$ are distinct for distinct linear orderings $S,S^{\mathrm{\prime }}$ and for some type $\tau$. In this paper, we will concentrate on ordinals and types of the form ${\sigma }_{a,b}=\mathrm{11...133...322...2}$. We will show that if $G(\alpha ,{\sigma }_{a,b})\cong G(\beta ,{\sigma }_{a,b})$ then $\alpha =\beta$. We will also consider the chromatic number and the automorphism groups of these graphs in order to gain a deeper understanding of their properties.