Abstract. Function clones are sets of functions on a fixed domain that are closed under composition and contain the projections. They carry a natural algebraic structure, provided by the laws of composition which hold in them, as well as a natural topological structure, provided by the topology of pointwise convergence, under which composition of functions becomes continuous. Inspired by recent results indicating the importance of the topological ego of function clones even for originally algebraic problems, we study questions of the following type: In which situations does the algebraic structure of a function clone determine its topological structure? We pay particular attention to function clones which contain an oligomorphic permutation group, and discuss applications of this situation in model theory and theoretical computer science.

It is known that a countable ω-categorical structure interprets all finite structures primitively positively if and only if its polymorphism clone maps to the clone of projections on a two-element set via a continuous clone homomorphism. We investigate the relationship between the existence of a clone homomorphism to the projection clone, and the existence of such a homomorphism which is continuous and thus meets the above criterion.

For n ≥ 3, let (Hn, E) denote the n-th Henson graph, i.e., the unique countable homogeneous graph with exactly those finite graphs as induced subgraphs that do not embed the complete graph on n vertices. We show that for all structures Γ with domain Hn whose relations are first-order definable in (Hn, E) the constraint satisfaction problem for Γ is either in P or is NP-complete.We moreover show a similar complexity dichotomy for all structures whose relations are first-order definable in a homogeneous graph whose reflexive closure is an equivalence relation.Together with earlier results, in particular for the random graph, this completes the complexity classification of constraint satisfaction problems of structures first-order definable in countably infinite homogeneous graphs: all such problems are either in P or NP-complete.An extended abstract of this paper has appeared at the 43rd International Colloquium on Automata, Languages and Programming (ICALP) Track B, 2016. and sinks [HN90, BKN09]). Various methods, combinatorial (graph-theoretic), logical, and universal-algebraic were brought to bear on this classification project, with many remarkable consequences. A conjectured delineation for the dichotomy was given in the algebraic language in [BKJ05], and finally the conjecture, and in particular this delineation, has recently been proven to be accurate [Bul17,Zhu17].When the domain is infinite, the complexity of the CSP can be outside NP, and even undecidable [BN06]. But for natural classes of such CSPs there is often the potential for structured classifications, and this has proved to be the case for structures first-order definable over the order (Q, <) of the rationals [BK09] or over the integers with successor [BMM17]. Another classification of this type has been obtained for CSPs where the constraint language is first-order definable over the random (Rado) graph [BP15a], making use of structural Ramsey theory. This paper was titled 'Schaefer's theorem for graphs' and it can be seen as lifting the famous classification of Schaefer [Sch78] from Boolean logic to logic over finite graphs, since the random graph is universal for the class of finite graphs.1.2. Homogeneous graphs and their reducts. The notion of homogeneity from model theory plays an important role when applying techniques from finite-domain constraint satisfaction to constraint satisfaction over infinite domains. A relational structure is homogeneous if every isomorphism between finite induced substructures can be extended to an automorphism of the entire structure. Homogeneous structures are uniquely (up to isomorphism) given by the class of finite structures that embed into them. The structure (Q, <) and the random graph are among the most prominent examples of homogeneous structures. The class of structures that are first-order definable over a homogeneous structure with finite relational signature is a very large generalization of the class of all finite structures, and CSPs for those structures have been studied independently in many dif...

Let a height function f be a real valued function on R 2 . A connected subset of R 2 is called an island, if there is a water level such that H is an island in the classical sense. We show that an island system is always laminar. Among others, in this paper we prove that the cardinality of a maximal laminar system is either countable or continuum.

The random ordered graph is the up to isomorphism unique countable homogeneous linearly ordered graph that embeds all finite linearly ordered graphs. We determine the reducts of the random ordered graph up to first-order interdefinability.