Results on the Generalised Shift Graph

In the paper ‘On Chromatic Number of Infinite Graphs’ (1968), Erdős and Hajnal defined the Shift Graph to be the graph whose vertices are the n-element subsets of some totally ordered set S, regarded as increasing n-tuples, such that A = (a1, ..., an) and B = (b1, ..., bn) are neighbours iff a1 < b1 = a2 <b2 = a3 < ... < bn−1 = an < bn or the other way round. In the paper ‘On Generalised Shift Graphs’ (2014), Avart, Łuczac and Rödl extend this definition to include all possible arrangements of the ais and bis, known as ‘types’. In this thesis, we will consider a selection of these types and study the corresponding graphs. All the types we consider will be written as 1^k3^m2^k, where k + m = n, which means that the final m entries of (a1, ..., an) are identified with the first m entries of (b1, ..., bn). Such a graph with totally ordered set S and type 1^k3^m2^k is denoted G(S,1^k3^m2^k). There are two related questions here. One is when the (undirected) graphs G(S,1^k3^m2^k) and G(S',1^k3^m2^k) are distinct (non-isomorphic) for distinct linear orderings S, S'. The other is to what extent we can recognise S inside the graph (called ‘reconstruction’). A positive solution to the latter also yields one for the former, since if we can recognise S in its graph, and S′ in its graph, and they are distinct, then so must the graphs be. We focus on these main cases: S is finite, S is an ordinal, S is a more general totally ordered set. The tools available for reconstruction depend on whether S is a total ordering, a dense total ordering, or an ordinal. There are additional technical complications in the case where S has endpoints, and similarly for S containing relatively small finite segments. Since these graphs are undirected, we expect in general only to recover a linear ordering up to order reversal. The natural notion here is of ‘linear betweenness’, and we spend some time studying linear betweenness relations in their own right, also considering the induced relations on n-tuples. Betweenness relations on n-tuples correspond to shift graphs of the special form G(S,1^n2^n) (i.e. in which no identifications are made). The main contribution of the thesis is to show how it is possible in many instances to reconstruct the underlying linear order (often just up to order-reversal) from the generalized shift graph. A typical example of this is Theorem 4.4. The techniques are to employ graph-theoretical features of the relevant shift graph, such as co-cliques or pairs of co-cliques fulfilling various conditions to ‘recognize’ points and relations of the underlying linear order. There are many variants depending on the precise circumstances (dense or not, with or without endpoints, well-ordered, only partially ordered). We show that for ordinals α and β, if G(α,1^k3^m2^k) is isomorphic to G(β,1^k3^m2^k) then α = β. Note that the fact that (in the infinite case) α is not isomorphic to its reversed ordering means that the betweenness relation is enough to give us the ordering. This result does not necessarily extend to all total orderings in full generality, but we obtain many results. A suite of techniques is used, which may be adapted suitably depending on circumstances, endpoints or not, density, or finiteness. In a more open-ended chapter, we generalise as much of the material for total orders to partial orders, the easiest case being that of trees. Work by Rubin [15] considers reconstruction in a slightly different sense: that a structure can be reconstructed from its automorphism group. So we have two ‘levels’ of reconstruction: of the graph from its automorphism group, and then if possible of the underlying total order from the graph. With this in mind, we study the automorphism groups of many of the graphs arising, managing in several cases to give quite explicit descriptions, so answering Rubin’s reconstruction question - i.e. whether or not a structure can be ‘re- constructed’ from its automorphism group (as in for example [17]) - where possible. For instance, we show that it is possible to determine S from Aut(G(S,132)) if and only if G(S, 132) contains no two points sharing exactly the same neighbour sets. Finally we return to colouring questions as in the original paper of Erdős and Hajnal, and show that the chromatic number of G(κ, 132) is equal to κ for any strong limit cardinal κ.