Miodrag Sokic

Let K be a class of finite relational structures. We denote by E_{n}K the class of structures obtained by adding equivalence relation with classes of size n. We denote by KE_{n} the class of structures obtained by blowing up points to classes of size n. We consider the Ramsey property for the order expansion of the classes E_{n}K and KE_{n}.

Let M be the class of finite metric spaces with non-zero distances in the set {1,2,3} omitting triangles of the form (2,3,3) and (3,3,3). We prove that the class of linearly ordered structures from M satisfies the Ramsey property.

Let Kn be the class of nite structures of the form (A;fA 1 ;:::;fA n ) where each fA i is a unary function on A. Let CKn be a class of nite structures of the form (A;fA 1 ;:::;fA n ;A) with the property that (A;fA 1 ;:::;fA n ) 2Kn and A is a linear ordering on A. We give an example of the Ramsey class CKn which satises the ordering property with respect to Kn.

We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable (in their natural topologies). For those which are amenable, we determine whether they are uniquely ergodic, leaving unsettled precisely one case (the \semi-generic" complete multipartite directed graph). We also consider the Hrushovski property. For most of our results we use the various techniques of [3], suitably generalized to a context in which the universal minimal flow is not necessarily the space of all orders. Negative results concerning amenability rely on constructions of the type considered in [26]. An additional class of structures (compositions) may be handled directly on the basis of very general principles. The starting point in all cases is the determination of the universal minimal flow for the automorphism group, which in the context of countable homogeneous directed graphs is given in [10] and the papers cited therein.

We study in this paper ordered nite measure algebras from the point of view of Frasse and Ramsey theory. We also propose an open problem, which is a homogeneous version of the Dual Ramsey Theorem of Graham-Rothschild, and derive consequences of a positive answer to the study of the topological dynamics of the automorphism group of a standard probability space and also the group of measure preserving homeomorphisms of the Cantor space.

We consider F, the class of Önite unary functions, i.e. the class of pairs of the form (A; f) where A is a Önite set and f is an unary function on A. We also consider two subclasses of F: B and Fk for k > 1. B contains structures (A; f) with the property that f is a bijection of the set A, and Fk contains structures (A; f) with the property that inverse image of every point in A under f has cardinality at most k. In this paper we calculate Ramsey degrees of structures from F and Fk, and we show that B is a Ramsey class. Moreover, we introduce various ordered expansions of the classes F and B and we prove Ramsey property for these expansions. In particular we prove Ramsey property for the class OF which contains structures of the form (A; f; ) where (A; f) 2 F and  is a linear ordering on the set A. In the case of the class Fk we introduce a precompact expansion with the Ramsey property. We also consider a generalization MnF, n > 1, of the class F which contains Önite structures of the form (A; f1; :::; fn) where each fi is an unary function on the set A. Finally we give a topological interpretation of our results by expanding the list of extremely amenable groups and by calculating various universal minimal flows.

 Let K be a class of finite relational structures. Let Kn be the class obtained by adding labeled partition, in at most n many parts, to structures in K. Let EK be the class obtained by adding unlabeled partition, i.e. an equivalence relation, to structures in K. We calculate Ramsey degrees for structures in Kn and EK. In addition, we apply our results to dynamics of minimal flows

We consider S, the class of finite semilattices; T, the class of finite treeable semilattices; and Tm, the subclass of T which contains trees with branching bounded by m. We prove that ES, the class of Önite lattices with linear extensions, is a Ramsey class. We calculate Ramsey degrees for structures in S, T and Tm. In addition to this we give a topological interpretation of our results and we apply our result to a canonization of linear orderings on finite semilattices. In particular, we give Örst two examples of a Fraisse class K which is not a Hrushovski class, and automorphism group of the Fraisse limit of K is non-trivial and it is uniquely ergodic.

Let L1 and L2 be two disjoint relational signatures. Let K1 and K2 be Ramsey classes of rigid relational structures in L1 and L2 respectively. Let K1⁎K2 be the class of structures in L1∪L2 whose reducts to L1 and L2 belong to K1 and K2 respectively. We give a condition on K1 and K2 which implies that K1⁎K2 is a Ramsey class. This is an extension of a result of M. Bodirsky.

In the second part of this paper we consider classes OS(2), OS(3), OB and OHwhich are obtained by expanding the class of finite dense local orders, the class of finite circular directed graphs, the class of finite boron tree structures, and the class of rooted trees respectively with linear orderings. We calculate Ramsey degrees for objects in these classes.

We introduce two Ramsey classes of finite relational structures. The first class contains finite structures of the form where \leq is a total ordering on A and % \preceq _{i} is a linear ordering on the set\{a\in A:I_{i}(a)\}. The second class contains structures of the form (A,\leq ,(I_{i})_{i=1}^{n},\preceq ) where (A,\leq ) is a weak ordering and % \preceq is a linear ordering on A such that A is partitioned by % \{a\in A:I_{i}(a)\} into maximal chains in the partial ordering \leq and each \{a\in A:I_{i}(a)\} is an interval with respect to \preceq .

 Let K be a class of finite relational structures. We define EK to be the class of finite relational structures A such that A/E∈K, where E is an equivalence relation defined on the structure A. Adding arbitrary linear orderings to structures from EK, we get the class OEK. If we add linear orderings to structures from EK such that each E-equivalence class is an interval then we get the class CE[K∗]. We provide a list of Fraïssé classes among EK, OEK and CE[K∗]. In addition, we classify OEK and CE[K∗] according to the Ramsey property. We also conduct the same analysis after adding additional structure to each equivalence class. As an application, we give a topological interpretation using the technique introduced in Kechris, Pestov and Todorčević. In particular, we extend the lists of known extremely amenable groups and universal minimal flows.

We introduce the class COU S of finite ultrametric spaces with distances in the set S and with two additional linear orderings. We also introduce the class EOP of finite posets with two additional linear orderings. In this paper, we prove that COU S and EOP are Ramsey classes. In addition, we give an application of our results to calculus of universal minimal flows.

A method is developed for proving non-amenability of certain automorphism groups of countable structures and is used to show that the automorphism groups of the random poset and random distributive lattice are not amenable. The universal minimal flow of the automorphism group of the random distributive lattice is computed as a canonical space of linear orderings but it is also shown that the class of finite distributive lattices does not admit hereditary order expansions with the Amalgamation Property.

We classify Fra¨ıss´e classes of finite posets with convex linear orderings with respect to the Ramsey property and extend the list of extremely amenable groups and universal minimal flows thanks to a theory developed by A. S. Kechris, V. G. Pestov and S. Todorˇcevi´c [Geom. Funct. Anal. 15 (2005), no. 1, 106–189; MR2140630 (2007j:37013)]. For the structures from the Schmerl list for which this technique is not applicable, we provide a direct calculation of universal minimal flows.

An important problem in topological dynamics is the calculation of the universal minimal flow of a topological group. When the universal minimal flow is one point, we say that the group is extremely amenable. For the automorphism group of Fraïssé structures, this problem has been translated into a question about the Ramsey and ordering properties of certain classes of finite structures by Kechris et al. (Geom Funct Anal 15:106–189, 2005). Using the Schmerl list (Schmerl, Algebra Univers 9:317–321, 1979) of Fraïssé posets, we consider classes of finite posets with arbitrary linear orderings and linear orderings that are linear extensions of the partial ordering. We provide classification of each of these classes according to their Ramsey and ordering properties. Additionally, we extend the list of extremely amenable groups as well as the list of metrizable universal minimal flows

We prove a finitary version of the Halpern–Läuchli Theorem. We also prove partition results about strong subtrees. Both results give estimates on the height of trees.