A homogenizable structure M is a structure where we may add a finite amount of new relational symbols to represent some 0-definable relations in order to make the structure homogeneous. In this article we will divide the homogenizable structures into different classes which categorize many known examples and show what makes each class important. We will show that model completeness is vital for the relation between a structure and the amalgamation bases of its age and give a necessary and sufficient condition for an countably categorical model-complete structure to be homogenizable.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

>k-homogeneous infinite graphs2018In: Journal of combinatorial theory. Series B (Print), ISSN 0095-8956, E-ISSN 1096-0902, Vol. 128, p. 160-174Article in journal (Refereed)

Abstract [en]

In this article we give an explicit classification for the countably infinite graphs G which are, for some k, ≥k-homogeneous. It turns out that a ≥k -homogeneous graph M is non-homogeneous if and only if it is either not 1-homogeneous or not 2-homogeneous, both cases which may be classified using ramsey theory.

This thesis is in the field of mathematical logic and especially model theory. The thesis contain six papers where the common theme is the Rado graph R. Some of the interesting abstract properties of R are that it is simple, homogeneous (and thus countably categorical), has SU-rank 1 and trivial dependence. The Rado graph is possible to generate in a probabilistic way. If we let K be the set of all finite graphs then we obtain R as the structure which satisfy all properties which hold with assymptotic probability 1 in K. On the other hand, since the Rado graph is homogeneous, it is also possible to generate it as a Fraïssé-limit of its age.

Paper I studies the binary structures which are simple, countably categorical, with SU-rank 1 and trivial algebraic closure. The main theorem shows that these structures are all possible to generate using a similar probabilistic method which is used to generate the Rado graph. Paper II looks at the simple homogeneous structures in general and give certain technical results on the subsets of SU-rank 1.

Paper III considers the set K consisting of all colourable structures with a definable pregeometry and shows that there is a 0-1 law and almost surely a unique definable colouring. When generating the Rado graph we almost surely have only rigid structures in K. Paper IV studies what happens if the structures in K are only the non-rigid finite structures. We deduce that the limit structures essentially try to stay as rigid as possible, given the restriction, and that we in general get a limit law but not a 0-1 law.

Paper V looks at the Rado graph's close cousin the random t-partite graph and notices that this structure is not homogeneous but almost homogeneous. Rather we may just add a definable binary predicate, which hold for any two elemenets which are in the same part, in order to make it homogeneous. This property is called being homogenizable and in Paper V we do a general study of homogenizable structures. Paper VI conducts a special case study of the homogenizable graphs which are the closest to being homogeneous, providing an explicit classification of these graphs.

In this article we give a classification of the binary, simple, ω-categorical structures with SU-rank 1 and trivial algebraic closure. This is done both by showing that they satisfy certain extension properties, but also by noting that they may be approximated by the almost sure theory of some sets of finite structures equipped with a probability measure. This study give results about general almost sure theories, but also considers certain attributes which, if they are almost surely true, generate almost sure theories with very specific properties such as ω-stability or strong minimality.

A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the kth level of the hierarchy can consist of all structures having at least k elements which are moved by some automorphism. Or we can consider, for any finite group G, all finite structures M such that G is a subgroup of the group of automorphisms of M; in this case the "hierarchy" is a partial order. In both cases, as well as variants of them, each "level" satisfies a logical limit law, but not a zero-one law (unless k = 0 or G is trivial). Moreover, the number of (labelled or unlabelled) n-element structures in one place of the hierarchy divided by the number of n-element structures in another place always converges to a rational number or to infinity as n -> infinity. All instances of the respective result are proved by an essentially uniform argument.

We study finite -colourable structures with an underlying pregeometry. The probability measure that is usedcorresponds to a process of generating such structures by which colours are first randomly assigned to all1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions aresatisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure,where we now forget the specific colouring of the generating process, has a given property. With this measurewe get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has anexplicit axiomatisation which is presented. (3) There is a formula ξ (x, y) (not directly speaking about colours)such that, with asymptotic probability 1, the relation “there is an -colouring which assigns the same colourto x and y” is defined by ξ (x, y). (4) With asymptotic probability 1, an -colourable structure has a unique-colouring (up to permutation of the colours).