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R.: Countable ultrahomogeneous undirected graphs
 Trans. AMS 262
, 1980
"... Abstract. Let G = (V0, £c> be an undirected graph. The complementary graph G is where (K „ V ¿ e Eô iff Vx + V2 and (K „ V ¿ C EG. Let K(n) be the complete undirected graph on n vertices and let £ be the graph i.e. <{a, b, c), {(b, c), (c, b)}}. G is ultrahomogeneous just in case every isomo ..."
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Cited by 56 (0 self)
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Abstract. Let G = (V0, £c> be an undirected graph. The complementary graph G is <KC,£ö> where (K „ V ¿ e Eô iff Vx + V2 and (K „ V ¿ C EG. Let K(n) be the complete undirected graph on n vertices and let £ be the graph i.e. <{a, b, c), {(b, c), (c, b)}}. G is ultrahomogeneous just in case every isomorphism of subgraph of smaller cardinality can be lifted to an automorphism of G. Let <3) = [K(n): n e u} u {E, É} u (K(n): n e <o}. Theorem: Le7 G „ G2 ¿>e fwo countable (infinite) ultrahomogeneous graphs such that for each H S <9 H can be embedded in G, just in case it can be embedded in G2. Then Gx a ¡ G2. Corollary: There are a countable number of countable ultrahomogeneous (undirected) graphs. 0. Introduction and preliminaries. A graph G is a pair <G, Ec) where \G\is the underlying or vertex set and EG is a binary relation on  G  called the edge set. A graph G is undirected just in case EG is symmetric and irreflexive. Where no confusion is likely to arise we make no distinction between G and  G . If X is a set by  X  we denote the cardinality of X. Countable means countable and infinite.
Step by Step  Building Representations in Algebraic Logic
 Journal of Symbolic Logic
, 1995
"... We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defini ..."
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Cited by 28 (15 self)
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We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is !categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...
Extending Partial Isomorphisms for the Small Index Property of Many OmegaCategorical Structures
 Israel Journal of Mathematics
, 1997
"... Theorem: Let A be a finite Km free graph, p 1 ; : : : ; pn partial isomorphisms on A. Then there exists a finite extension B, which is also a Km free graph, and automorphisms f i of B extending the p i . A paper by Hodges, Hodkinson, Lascar and Shelah shows how this theorem can be used to prove ..."
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Cited by 18 (2 self)
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Theorem: Let A be a finite Km free graph, p 1 ; : : : ; pn partial isomorphisms on A. Then there exists a finite extension B, which is also a Km free graph, and automorphisms f i of B extending the p i . A paper by Hodges, Hodkinson, Lascar and Shelah shows how this theorem can be used to prove the small index property for the generic countable graph of this class. The same method also works for a certain class of continuum many nonisomorphic !categorical countable digraphs and more generally for structures in an arbitrary finite relational language, which are built in a similar fashion. Hrushovski proved this theorem for the class of all finite graphs [Hr]; the proof presented here stems from his proof. 1. Introduction We say a class K of structures has the extension property for partial isomorphisms, (EP) for short, if for every finite structure A2K and p 1 ; : : : ; pn partial isomorphisms on A, there exists a finite structure B2K and f 1 ; : : : ; fn 2Aut(B) such that f...
A new uncountably categorical group
 Trans. Amer. Math. Soc
, 1996
"... Abstract. We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CMtrivial. 1. ..."
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Cited by 17 (3 self)
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Abstract. We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CMtrivial. 1.
Computational Complexity of Determining Which Statements about Causality Hold
 in Different SpaceTime Models”, Theoretical Computer Science
"... Causality is one of the most fundamental notions of physics. It is therefore important to be able to decide which statements about causality are correct in different models of spacetime. In this paper, we analyze the computational complexity of the corresponding deciding problems. In particular, we ..."
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Cited by 10 (7 self)
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Causality is one of the most fundamental notions of physics. It is therefore important to be able to decide which statements about causality are correct in different models of spacetime. In this paper, we analyze the computational complexity of the corresponding deciding problems. In particular, we show that: • for Minkowski spacetime, the deciding problem is as difficult as the Tarski’s problem of deciding elementary geometry, while • for a natural model of primordial spacetime, the corresponding deciding problem is of the lowest possible complexity. 1
Geometric grid classes of permutations
 Trans. Amer. Math. Soc
"... A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ±1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forb ..."
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Cited by 9 (8 self)
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A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ±1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.
SMALL PERMUTATION CLASSES
"... We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number κ, approximately 2.20557, for which there are only countably many permutation classes of growth rate (StanleyWilf limit) less than κ but uncountab ..."
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Cited by 6 (2 self)
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We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number κ, approximately 2.20557, for which there are only countably many permutation classes of growth rate (StanleyWilf limit) less than κ but uncountably many permutation classes of growth rate κ, answering a question of Klazar. We go on to completely characterize the possible subκ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced herein), partial wellorder, and atomicity (also known as the joint embedding property). 1.
From the Coxeter graph to the Klein graph
"... We show that the 56vertex Klein cubic graph Γ ′ can be obtained from the 28vertex Coxeter cubic graph Γ by ’zipping ’ adequately the squares of the 24 7cycles of Γ endowed with an orientation obtained by considering Γ as a Cultrahomogeneous digraph, where C is the collection formed by both the o ..."
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Cited by 5 (5 self)
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We show that the 56vertex Klein cubic graph Γ ′ can be obtained from the 28vertex Coxeter cubic graph Γ by ’zipping ’ adequately the squares of the 24 7cycles of Γ endowed with an orientation obtained by considering Γ as a Cultrahomogeneous digraph, where C is the collection formed by both the oriented 7cycles ⃗ C7 and the 2arcs ⃗ P3 that tightly fasten those ⃗ C7 in Γ. In the process, it is seen that Γ ′ is a C ′ultrahomogeneous (undirected) graph, where C ′ is the collection formed by both the 7cycles C7 and the 1paths P2 that tightly fasten those C7 in Γ ′. This yields an embedding of Γ ′ into a 3torus T3 which forms the Klein map of Coxeter notation (7, 3)8. The dual graph of Γ ′ in T3 is the distanceregular Klein quartic graph, with corresponding dual map of Coxeter notation (3,7)8.
Overview of some general results in combinatorial enumeration
, 2008
"... This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part ..."
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Cited by 5 (0 self)
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This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of contextfree languages, 3. holonomicity (i.e., Precursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn −3/2 r n and 5. ultimate modular periodicity of numbers of MSOLdefinable structures. 1