All reducts of the random graph are model-complete

We study locally closed transformation monoids which contain the automorphism group of the random graph. We show that such a transformation monoid is locally generated by the permutations in the monoid, or contains a constant operation, or contains an operation that maps the random graph injectively to an induced subgraph which is a clique or an independent set. As a corollary, our techniques yield a new proof of Simon Thomas' classification of the five closed supergroups of the automorphism group of the random graph; our proof uses different Ramsey-theoretic tools than the one given by Thomas, and is perhaps more straightforward. Since the monoids under consideration are endomorphism monoids of relational structures definable in the random graph, we are able to draw several model-theoretic corollaries: One consequence of our result is that all structures with a first-order definition in the random graph are model-complete. Moreover, we obtain a classification of these structures up to existential interdefinability.Comment: Technical report not intended for publication in a journal. Subsumed by the more recent article 1003.4030. Length 14 pages

The Reducts of the Homogeneous Binary Branching C-relation

Let (L;C) be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of (L;C), i.e., the structures with domain L that are first-order definable in (L;C). We show that up to existential interdefinability, there are finitely many such reducts. This implies that there are finitely many reducts up to first-order interdefinability, thus confirming a conjecture of Simon Thomas for the special case of (L;C). We also study the endomorphism monoids of such reducts and show that they fall into four categories.Comment: 39 pages, 4 figure

Generalised Indiscernibles, Dividing Lines, and Products of Structures

Generalised indiscernibles highlight a strong link between model theory and structural Ramsey theory. In this paper, we use generalised indiscernibles as tools to prove results in both these areas. More precisely, we first show that a reduct of an ultrahomogenous $\aleph_0$-categorical structure which has higher arity than the original structure cannot be Ramsey. In particular, the only nontrivial Ramsey reduct of the generically ordered random $k$-hypergraph is the linear order. We then turn our attention to model-theoretic dividing lines that are characterised by collapsing generalised indiscernibles, and prove, for these dividing lines, several transfer principles in (full and lexicographic) products of structures. As an application, we construct new algorithmically tame classes of graphs

Homogeneous Structures: Model Theory meets Universal Algebra (online meeting)

The workshop "Homogeneous Structures: Model Theory meets Universal Algebra'' was centred around transferring recently obtained advances in universal algebra from the finite to the infinite. As it turns out, the notion of homogeneity together with other model-theoretic concepts like $\omega$-categoricity and the Ramsey property play an indispensable role in this endeavour