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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 <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 ..."
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Cited by 84 (1 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.
Structure of partially ordered sets with transitive automorphism groups
 Mem. Amer. Math. Soc
, 1985
"... In this paper, we study the structure of infinite partially ordered sets (Q, ==) under suitable transitivity assumptions on their group A(£l) of all orderautomorphisms of (Q, =s). Let us call A(Q) ifetransitive (Jfchomogeneous) if whenever A, B are two isomorphic subsets of Q each with k elements ..."
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Cited by 15 (0 self)
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In this paper, we study the structure of infinite partially ordered sets (Q, ==) under suitable transitivity assumptions on their group A(£l) of all orderautomorphisms of (Q, =s). Let us call A(Q) ifetransitive (Jfchomogeneous) if whenever A, B are two isomorphic subsets of Q each with k elements, then some (any) isomorphism from (A, =£) onto (JB, «s) extends to an automorphism of Q, respectively. We show that if it & 4 (k = 3), there are precisely k (5) nonisomorphic countable partially ordered sets (Q, =£) not containing the pentagon such that A(Q) is A:transitive but not ^homogeneous; if k = 2, there are a unique countable, and many different uncountable sets (Q, «s) of this type. We also give necessary and sufficient conditions for two partially ordered sets (Q, =s) not containing the pentagon and with Jttransitive automorphism group (fc>2) to be L^equivalent. 1. Introduction and
On a {K4, K2,2,2}ultrahomogeneous graph
 Australasian Journal of Combinatorics
"... The existence of a connected 12regular {K4, K2,2,2}ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way ..."
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Cited by 14 (14 self)
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The existence of a connected 12regular {K4, K2,2,2}ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way both as the edgedisjoint union of 42 induced copies of K4 and as the edgedisjoint union of 21 induced copies of K2,2,2, with no more copies of K4 or K2,2,2 existing in G. Moreover, each edge of G is shared by exactly one copy of K4 and one of K2,2,2. While the line graphs of dcubes, (3 ≤ d ∈ ZZ), are {Kd, K2,2}ultrahomogeneous, G is not even linegraphical. In addition, the chordless 6cycles of G are seen to play an interesting role and some selfdual configurations associated to G with 2arctransitive, arctransitive and semisymmetric Levi graphs are considered. 1
Some Aspects of Model Theory and Finite Structures
, 2002
"... this paper is to highlight some of these aspects of the model theory of nite structures, where the nite and in nite interact fruitfully, in order to dispel the perhaps too common perception that ( rstorder) model theory has little to say about nite structures ..."
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Cited by 11 (0 self)
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this paper is to highlight some of these aspects of the model theory of nite structures, where the nite and in nite interact fruitfully, in order to dispel the perhaps too common perception that ( rstorder) model theory has little to say about nite structures
Distance transitive graphs and finite simple groups
, 1987
"... This paper represents the first step in the classification of finite primitive distance transitive graphs. In it we reduce the problem to the case where the automorphism group is either almost simple or affine. Let ^ be a simple, connected, undirected graph with vertex set Q. If oc, /? e Q, ..."
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Cited by 9 (4 self)
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This paper represents the first step in the classification of finite primitive distance transitive graphs. In it we reduce the problem to the case where the automorphism group is either almost simple or affine. Let ^ be a simple, connected, undirected graph with vertex set Q. If oc, /? e Q,
HomomorphismHomogeneous Graphs
"... We answer two open questions posed by Cameron and Nesetril concerning homomorphismhomogeneous graphs. In particular we show, by giving a characterization of these graphs, that extendability to monomorphism or to homomorphism leads to the same class of graphs when defining homomorphismhomogeneity. F ..."
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We answer two open questions posed by Cameron and Nesetril concerning homomorphismhomogeneous graphs. In particular we show, by giving a characterization of these graphs, that extendability to monomorphism or to homomorphism leads to the same class of graphs when defining homomorphismhomogeneity. Further we show that there are homomorphismhomogeneous graphs that do not contain the Rado graph as a spanning subgraph answering the second open question. We also treat the case of homomorphismhomogeneous graphs with loops allowed, showing that the corresponding decision problem is coNP complete. Finally we extend the list of considered morphismtypes and show that the graphs for which monomorphisms can be extended to epimorphisms are complements of homomorphism homogeneous graphs. 1
COUNTABLE CONNECTEDHOMOGENEOUS GRAPHS
"... A graph is connectedhomogeneous if any isomorphism between finite connected induced subgraphs extends to an automorphism of the graph. In this paper we classify the countably infinite connectedhomogeneous graphs. We prove that if Γ is connected countably infinite and connectedhomogeneous then Γ i ..."
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Cited by 7 (0 self)
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A graph is connectedhomogeneous if any isomorphism between finite connected induced subgraphs extends to an automorphism of the graph. In this paper we classify the countably infinite connectedhomogeneous graphs. We prove that if Γ is connected countably infinite and connectedhomogeneous then Γ is isomorphic to one of: Lachlan and Woodrow’s ultrahomogeneous graphs; the generic bipartite graph; the bipartite ‘complement of a complete matching’; the line graph of the complete bipartite graph Kℵ0,ℵ0; or one of the ‘treelike ’ distancetransitive graphs Xκ1,κ2 where κ1, κ2 ∈ N∪{ℵ0}. It then follows that an arbitrary countably infinite connectedhomogeneous graph is a disjoint union of a finite or countable number of disjoint copies of one of these graphs, or to the disjoint union of countably many copies of a finite connectedhomogeneous graph. The latter were classified by Gardiner (1976). We also classify the countably infinite connectedhomogeneous posets.
What is a Structure Theory
 Bulletin of the London Mathematical Society
, 1987
"... into the 'Mathematical logic and foundations ' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consis ..."
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Cited by 6 (0 self)
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into the 'Mathematical logic and foundations ' section of the June 1982 issue must have been puzzled by one of the items there. It was submitted by Saharon Shelah of the Hebrew University at Jerusalem, and its title was 'Why am I so happy?'. The text of the abstract mainly consisted of equalities and inequalities between uncountable cardinals. In my experience most mathematicians find uncountable cardinals depressing, if they have any reaction to them at all. In fact Shelah was quite right to be so happy, but not because of his cardinal inequalities. He had just brought to a successful conclusion a line of research which had cost him fourteen years of intensive work and not far off a hundred published books and papers. In the course of this work he had established a new range of questions about mathematics with implications far beyond mathematical logic. That is what I want to discuss here. In brief, Shelah's work is about the notion of a class of structures which has a good structure theory. We all have a rough intuitive notion of what counts as a good structure theory. For example the structure theory of finitely generated abelian groups