We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Journal of the ACM

https://ambio1.leeds.ac.uk/Pure/staff/macpherson/homog7.pdf

A survey of homogeneous structures

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Hitchhiker's guide to the galaxy

A temporal constraint language is a set of relations that has a first-order definition in(Q;

/pdf/the-complexity-of-temporal-constraint-satisfaction-problems-5329j01r81.pdf

The complexity of temporal constraint satisfaction problems

A fundamental fact for the algebraic theory of constraint satisfaction problems (CSPs) over a fixed template is that pp-interpretations between at most countable ω-categorical relational structures have two algebraic counterparts for their polymorphism clones: a semantic one via the standard algebraic operators H, S, P, and a syntactic one via clone homomorphisms (capturing identities). We provide a similar characterization which incorporates all relational constructions relevant for CSPs, that is, homomorphic equivalence and adding singletons to cores in addition to ppinterpretations. For the semantic part we introduce a new construction, called reflection, and for the syntactic part we find an appropriate weakening of clone homomorphisms, called h1 clone homomorphisms (capturing identities of height 1). As a consequence, the complexity of the CSP of an at most countable ω-categorical structure depends only on the identities of height 1 satisfied in its polymorphism clone as well as the natural uniformity thereon. This allows us in turn to formulate a new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures. Finally, we reveal a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties.

/pdf/the-wonderland-of-reflections-3w3l5o1kw5.pdf

The wonderland of reflections

Schaefer's theorem is a complexity classification result for so-called Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NP-complete.We present an analog of this dichotomy result for the propositional logic of graphs instead of Boolean logic. In this generalization of Schaefer's result, the input consists of a set W of variables and a conjunction Φ of statements (“constraints”) about these variables in the language of graphs, where each statement is taken from a fixed finite set Ψ of allowed quantifier-free first-order formulas; the question is whether Φ is satisfiable in a graph.We prove that either Ψ is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time, or the problem is NP-complete. This is achieved by a universal-algebraic approach, which in turn allows us to use structural Ramsey theory. To apply the universal-algebraic approach, we formulate the computational problems under consideration as constraint satisfaction problems (CSPs) whose templates are first-order definable in the countably infinite random graph. Our method for classifying the computational complexity of those CSPs is based on a Ramsey-theoretic analysis of functions acting on the random graph, and we develop general tools suitable for such an analysis which are of independent mathematical interest.

Schaefer's Theorem for Graphs

One way of studying a relational structure is to investigate functions which are related to that structure and which leave certain aspects of the structure invariant. Examples are the automorphism group, the self-embedding monoid, the endomorphism monoid, or the polymorphism clone of a structure. Such functions can be particularly well understood when the relational structure is countably infinite and has a first-order definition in another relational structure which has a finite language, is totally ordered and homogeneous, and has the Ramsey property. This is because in this situation, Ramsey theory provides the combinatorial tool for analyzing these functions -- in a certain sense, it allows to represent such functions by functions on finite sets. 
This is a survey of results in model theory and theoretical computer science obtained recently by the authors in this context. In model theory, we approach the problem of classifying the reducts of countably infinite ordered homogeneous Ramsey structures in a finite language, and certain decidability questions connected with such reducts. In theoretical computer science, we use the same combinatorial methods in order to classify the computational complexity for various classes of infinite-domain constraint satisfaction problems. While the first set of applications is obviously of an infinitary character, the second set concerns genuinely finitary problems -- their unifying feature is that the same tools from Ramsey theory are used in their solution.

Reducts of Ramsey structures

We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain.

/pdf/the-reducts-of-equality-up-to-primitive-positive-4fihc60st6.pdf

The reducts of equality up to primitive positive interdefinability

For a fixed countably infinite structure $\Gamma$ with finite relational signature $\tau$, we study the following computational problem: input are quantifier-free $\tau$-formulas $\phi_0,\phi_1,\dots,\phi_n$ that define relations $R_0,R_1,\dots,R_n$ over $\Gamma$. The question is whether the relation $R_0$ is primitive positive definable from $R_1,\dots,R_n$, i.e., definable by a first-order formula that uses only relation symbols for $R_1, \dots, R_n$, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden). We show decidability of this problem for all structures $\Gamma$ that have a first-order definition in an ordered homogeneous structure $\Delta$ with a finite relational signature whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples of structures $\Gamma$ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal $C$-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.

Michael Pinsker

Papers

The wonderland of reflections

Schaefer's Theorem for Graphs

Reducts of Ramsey structures

The reducts of equality up to primitive positive interdefinability

Decidability of definability