Primitive groups, graph endomorphisms and synchronization

The third author has been partially supported by the Fundação para a Ciência e a Tecnologia through the project CEMAT-CIÊNCIAS UID/Multi/04621/2013.Let Ω be a set of cardinality n, G be a permutation group on Ω and f:Ω→Ω be a map that is not a permutation. We say that G synchronizes f if the transformation semigroup ⟨G,f⟩ contains a constant map, and that G is a synchronizing group if G synchronizes every non-permutation.  A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it had previously been conjectured that this was essentially the only way in which a primitive group could fail to be synchronizing, in other words, that a primitive group synchronizes every non-uniform transformation.  The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every non-uniform transformation of rank at most 4, these examples are of the lowest possible rank. In addition, we produce graphs with primitive automorphism groups that have approximately √n non-synchronizing ranks, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group. The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree n synchronizes every non-uniform transformation of rank n−1 and n−2, and here this is extended to n−3 and n−4.  In the process, we will obtain a purely graph-theoretical result showing that, with limited exceptions, in a vertex-primitive graph the union of neighbourhoods of a set of vertices A is bounded below by a function that is asymptotically √|A|. Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.PostprintPeer reviewe

Definable Sets in Finite Structures

This Thesis is primarily motivated by a conjecture of Anscombe, Macpherson, Steinhorn and Wolf [2]. The conjecture states that, for a homogeneous structure M over a finite relational language, M is elementarily equivalent to the ultraproduct of a ‘multidimensional exact class’ if and only if M is stable. The right to left statement has already been verified, and so our focus is on the left to right. In this thesis, we confirm the conjecture for certain unstable homogeneous structures such as the universal metrically homogeneous graph of diameter k, the universal homogeneous two-graph and various others, such as the 28 ‘semi- free’ edge-coloured homogeneous graphs described by Cherlin in the appendix of [16]. We also provide some mechanisms for answering the question for other unstable structures. The core of this thesis is about finite ‘n-regular’ 3-edge-coloured graphs. For any given n, a classification of sufficiently large n-regular 3-edge-coloured graphs is expected to yield a proof of the ‘m.e.c’ conjecture in the case of the universal homogeneous 3-coloured graph, and indeed, our results yield some further special cases of the ‘m.e.c’ conjecture. The main focus is on finite ‘3-regular’ 3-coloured graphs. We classify such structures under certain conditions: when they possess a ‘complete neighbourhood’, when they are ‘monochromatic-triangle-free’ and if we increase to ‘4-regularity’ we can classify the imprimitive case as well. In the other scenarios, we employ methods from the theory of association schemes, together with linear algebra, to give a description of the eigenvalues and/or eigenvectors of the neighbourhoods with respect to a base point. We also describe the two known primitive examples of such graphs and prove they are actually homogeneous, which implies n-regularity for each n

Omega-categorical simple theories

This thesis touches on many different aspects of homogeneous relational structures. We start with an introductory chapter in which we present all the background from model theory and homogeneity necessary to understand the results in the main chapters. The second chapter is a list of examples. We present examples of binary and ternary homogeneous relational stuctures, and prove the simplicity or non-simplicity of their theory. Many of these examples are well-known structures (the ordered rational numbers, random graphs and hypergraphs, the homogeneous Kn-free graphs), while others were constructed during the first stages of research. In the same chapter, we present some combinatorial results, including a proof of the TP2 in the Fraïssé limit of semifree amalgamation classes in the language of n-graphs, such that all the minimal forbidden configurations of the class of size at least 3 are all triangles. The third chapter contains the main results of this thesis. We prove that supersimple finitely homogeneous binary relational structures cannot have infinite monomial SU-rank, show that primitive binary supersimple homogeneous structures of rank 1 are “random” in the sense that all their minimal forbidden configurations are of size at most 2, and partially classify the supersimple 3-graphs under the assumption of stable forking in the theories of finitely homogeneous structures with supersimple theory. The fourth chapter is a proof of the directed-graph version of a well-known result by Erdős, Kleitman and Rothschild. Erdős et al. prove that almost all finite labelled trianglefree simple graphs are bipartite, and we prove that almost all finite labelled directed graphs in which any three distinct vertices span at least one directed arc consist of two disjoint tournaments, possibly with some directed arcs from one to the other

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