On graph norms for complex-valued functions

For any given graph $H$, one may define a natural corresponding functional $\|.\|_H$ for real-valued functions by using homomorphism density. One may also extend this to complex-valued functions, once $H$ is paired with a $2$-edge-colouring $\alpha$ to assign conjugates. We say that $H$ is real-norming (resp. complex-norming) if $\|.\|_H$ (resp. $\|.\|_{H,\alpha}$ for some $\alpha$) is a norm on the vector space of real-valued (resp. complex-valued) functions. These generalise the Gowers octahedral norms, a widely used tool in extremal combinatorics to quantify quasirandomness. We unify these two seemingly different notions of graph norms in real- and complex-valued settings. Namely, we prove that $H$ is complex-norming if and only if it is real-norming and simply call the property norming. Our proof does not explicitly construct a suitable $2$-edge-colouring $\alpha$ but obtains its existence and uniqueness, which may be of independent interest. As an application, we give various example graphs that are not norming. In particular, we show that hypercubes are not norming, which resolves the last outstanding problem posed in Hatami's pioneering work on graph norms.Comment: 33 page

Diamond-free partial orders

This thesis presents initial work in attempting to understand the class of ‘diamond-free’ 3-cs-transitive partial orders. The notion of diamond-freeness, proposed by Gray, says that for any a ≤ b, the set of points between a and b is linearly ordered. A weak transitivity condition called ‘3-cs-transitivity’ is taken from the corresponding notion for cycle-free partial orders, which in that case led to a complete classification [3] of the countable examples. This says that the automorphism group acts transitively on certain isomorphism classes of connected 3-element structures. Classification for diamond-free partial orders seems at present too ambitious, but the strategy is to seek classifications of natural subclasses, and to test conjectures suggested by motivating examples. The body of the thesis is divided into three main inter-related chapters. The first of these, Chapter 3, adopts a topological approach, focussing on an analogue of topological covering maps. It is noted that the class of ‘covering projections’ between diamond-free partial orders can add symmetry or add cycles, and notions such as path connectedness transfer directly. The concept of the ‘nerve’ of a partial order makes this analogy concrete, and leads to useful observations about the fundamental group and the existence of an underlying cycle-free partial order called the universal cover. In Chapter 4, the work of [1] is generalised to show how to decompose ranked diamond- free partial orders. As in the previous chapter, any diamond-free partial order is covered by a specific cycle-free partial order. The paper [1] constructs a diamond-free partial order with cycles of height 1 from a different cycle-free partial order through which the universal covering factors. This is extended to construct a sequence of diamond- free partial orders with cycles of finite height which are not only factors but have the chosen diamond-free partial order as a ‘limit’. This leads to a better understanding of why structures with cycles only of height 1 are special, and the rest divide into structures with cycles of bounded height and a cycle-free backbone, and those for which the cycles have cofinal height. Even these can be expressed as limits of structures with cycles of 6 bounded height, though not directly. A variety of constructions are presented in Chapter 5, based on an underlying cycle- free partial order, and an ‘anomaly’, which in the simplest case given in [5] is a 2-level Dedekind-MacNeille complete 3-cs-transitive partial order, but which here is allowed to be a partial order of greater complexity. A rich class of examples is found, which have very high degrees of homogeneity and help to answer a number of conjectures in the negative

Homomorphisms of (j,k)-mixed graphs

A mixed graph is a simple graph in which a subset of the edges have been assigned directions to form arcs. For non-negative integers j and k, a (j,k)−mixed graph is a mixed graph with j types of arcs and k types of edges. The collection of (j,k)−mixed graphs contains simple graphs ((0,1)−mixed graphs), oriented graphs ((1,0)−mixed graphs) and k−edge- coloured graphs ((0,k)−mixed graphs).A homomorphism is a vertex mapping from one (j,k)−mixed graph to another in which edge type is preserved, and arc type and direction are preserved. The (j,k)−chromatic number of a (j,k)−mixed graph is the least m such that an m−colouring exists. When (j,k)=(0,1), we see that these definitions are consistent with the usual definitions of graph homomorphism and graph colouring.In this thesis we study the (j,k)−chromatic number and related parameters for different families of graphs, focussing particularly on the (1,0)−chromatic number, more commonly called the oriented chromatic number, and the (0,k)−chromatic number.In addition to considering vertex colourings, we also consider incidence colourings of both graphs and digraphs. Using systems of distinct representatives, we provide a new characterisation of the incidence chromatic number. We define the oriented incidence chromatic number and find, by way of digraph homomorphism, a connection between the oriented incidence chromatic number and the chromatic number of the underlying graph. This connection motivates our study of the oriented incidence chromatic number of symmetric complete digraphs.Un graphe mixte est un graphe simple tel que un sous-ensemble des arêtes a une orientation. Pour entiers non négatifs j et k, un graphe mixte-(j,k) est un graphe mixte avec j types des arcs and k types des arêtes. La famille de graphes mixte-(j,k) contient graphes simple, (graphes mixte−(0,1)), graphes orienté (graphes mixte−(1,0)) and graphe coloré arête −k (graphes mixte−(0,k)).Un homomorphisme est un application sommet entre graphes mixte−(j,k) que tel les types des arêtes sont conservés et les types des arcs et leurs directions sont conservés. Le nombre chromatique−(j,k) d’un graphe mixte−(j,k) est le moins entier m tel qu’il existe un homomorphisme à une cible avec m sommets. Quand on observe le cas de (j,k) = (0,1), on peut déterminer ces définitions correspondent à les définitions usuel pour les graphes.Dans ce mémoire on etude le nombre chromatique−(j,k) et des paramètres similaires pour diverses familles des graphes. Aussi on etude les coloration incidence pour graphes and digraphs. On utilise systèmes de représentants distincts et donne une nouvelle caractérisation du nombre chromatique incidence. On define le nombre chromatique incidence orienté et trouves un connexion entre le nombre chromatique incidence orienté et le nombre chromatic du graphe sous-jacent

Multicoloured Random Graphs: Constructions and Symmetry

This is a research monograph on constructions of and group actions on countable homogeneous graphs, concentrating particularly on the simple random graph and its edge-coloured variants. We study various aspects of the graphs, but the emphasis is on understanding those groups that are supported by these graphs together with links with other structures such as lattices, topologies and filters, rings and algebras, metric spaces, sets and models, Moufang loops and monoids. The large amount of background material included serves as an introduction to the theories that are used to produce the new results. The large number of references should help in making this a resource for anyone interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will appear in physic