Set-homogeneous hypergraphs

A $k$-uniform hypergraph $M$ is set-homogeneous if it is countable (possibly finite) and whenever two finite induced subhypergraphs $U,V$ are isomorphic there is $g\in Aut(M)$ with $U^g=V$; the hypergraph $M$ is said to be homogeneous if in addition every isomorphism between finite induced subhypergraphs extends to an automorphism. We give four examples of countably infinite set-homogeneous $k$-uniform hypergraphs which are not homogeneous (two with $k=3$, one with $k=4$, and one with $k=6$). Evidence is also given that these may be the only ones, up to complementation. For example, for $k=3$ there is just one countably infinite $k$-uniform hypergraph whose automorphism group is not 2-transitive, and there is none for $k=4$. We also give an example of a finite set-homogeneous 3-uniform hypergraph which is not homogeneous.


Introduction 1.Background
A very rich theory has developed around the notions of Fraïssé amalgamation and homogeneous structure in the sense of Fraïssé.At the most basic level the notions concern countable structures over finite relational languages, but the concepts make good sense for languages with function and constant symbols, and there are category-theoretic versions, versions with inverse rather than direct limits, and formulations of amalgamation with respect to a specified class of embeddings, leading to 'Hrushovski constructions'.The subject has important connections to permutation group theory, model theory, combinatorial enumeration, Ramsey theory, topological dynamics, and constraint satisfaction.In certain specified binary contexts, there are classification theorems of the homogeneous structures: for example for partial orders [36], coloured partial orders [38], graphs [29], digraphs [13], 'finite-dimensional permutation structures' (structures in a language with finitely many total order symbols) [6], and metrically homogeneous graphs of diameter 3 [3].However, there is currently very little (beyond [2]) in the way of classification theorems for homogeneous structures where relation symbols have arity greater than two, and evidence from binary classifications suggests such results will be very difficult.
In this paper we focus on k-uniform hypergraphs (from now on, called just k-hypergraphs) with k ≥ 3, and on a slight weakening of homogeneity.Let L be a relational language, and let t ∈ N with t ≥ 1.A relational structure M over L is t-set-homogeneous if |M | ≤ ℵ 0 (we assume this throughout) and, whenever U, V ⊂ M are of size t and carry isomorphic substructures, there is g ∈ Aut(M ) with U g = V ; it is ≤t-set-homogeneous if it is s-set-homogeneous for all s ≤ t.The structure M is set-homogeneous if it is t-set-homogeneous for all t ∈ N with t > 0. We say a countable structure M is t-homogeneous if any isomorphism between substructures of M of size t extends to an automorphism of M , and ≤t-homogeneous if it is s-homogeneous for all s ≤ t; the structure M is homogeneous if it is t-homogeneous for all t > 0. A t-set-homogeneous structure M is t-homogeneous if and only if, for each A ⊂ M of size t, the group induced on A be the setwise stabiliser of A in Aut(M ) is precisely the full automorphism group of the structure induced on A.
We are not sure of the history of set-homogeneity, but there is some discussion of the notion in Section 8 of [21,Chapter 11].We remark that Hall's universal locally finite group is often described as the unique countably infinite locally finite group which embeds all finite groups and has the property that any two finite isomorphic subgroups are conjugate.This is a set-homogeneity condition, but in this specific group-theoretic context it implies the stronger homogeneity condition: any isomorphism between finite subgroups of H is induced by some inner automorphism; see [26, Lemma 3].
Finite homogeneous graphs were classified independently by Gardiner [22] and by Golfand and Klin [23].It was shown by Ronse [35] that any finite set-homogeneous graph is homogeneous, and Enomoto [19] gave a very short direct proof of this.Enomoto's argument was shown in [25,Lemma 3.1] to work for finite tournaments, but not for finite digraphs (a directed 5-cycle is sethomogeneous but not homogeneous).Finite set-homogeneous directed graphs (allowing pairs with an arc in each direction) were classified in [25], and [40] initiates an analysis of finite 3-set-homogeneous graphs.
There is little literature so far on infinite set-homogeneous structures.Sethomogeneous graphs are considered in [18], where a specific countable infinite graph R(3) related to a circular order is shown to be set-homogeneous but not homogeneous, and it is shown that any countably infinite graph which is ≤8-set-homogeneous but not ≤3-homogeneous is isomorphic to R(3) or its complement.Likewise, in [25], a classification is given of countably infinite sethomogeneous digraphs (not allowing pairs with arcs in both directions) which are not 2-homogeneous.Some papers of Cameron on multiply-homogeneous permutation groups ( [7], [8], [9]) have similar flavour -see Section 2.2 for more on this.The paper [17] contains several classification results for countable partially ordered sets which are k-set-homogeneous (called k-transitive in [17]) but not k-homogeneous and do not contain the pentagon.In a similar spirit, [24] includes a classification of locally finite graphs with more than one end which are 3-CS-transitive (whenever U , V are connected induced subgraphs of size at most 3 which are isomorphic, there is an automorphism g with U g = V ).

The main results
In this paper, we construct four specific infinite hypergraphs and prove they are set-homogeneous.We then characterise some of these examples up to complementation by low degree transitivity and primitivity properties of their automorphism group.Before stating our main results, we recall that a permutation group G on a set X is k-homogeneous if it is transitive on the collection of unordered k-subsets of X, and is k-transitive if it is transitive on the ordered k-subsets.(Note a slight inconsistency which seems to be established in the literature; for structures, homogeneity is the stronger of the two conditions khomogeneity/k-set-homogeneity, but for permutation groups, it is the weaker of the conditions k-homogeneity/k-transitivity.)We say G is k-primitive on X if it is k-transitive and for any distinct x 1 , . . ., x k−1 ∈ X the stabiliser G x1...x k−1 acts primitively on X \ {x 1 , . . ., x k−1 }, that is, preserves no proper non-trivial equivalence relation.
Our main theorems are the following.We stress the assumption throughout that set-homogeneity conditions for us imply by definition that a structure is countable, so our structures are throughout assumed to be countable.

Theorem A. (i) There is an infinite set-homogeneous but not homogeneous
3-hypergraph M 3 whose automorphism group is not 2-transitive, and any infinite ≤4-set-homogeneous 3-hypergraph whose automorphism group is not 2-transitive is isomorphic to M 3 .
(iii) If M is an infinite ≤5-set-homogeneous 4-hypergraph such that Aut(M ) is not 2-primitive then either M is isomorphic to M 4 or its complement, or Aut(M ) preserves a linear betweenness relation on M .

Theorem C. (i)
There is an infinite set-homogeneous but not homogeneous 6-hypergraph M 6 whose automorphism group is 3-transitive but not 3primitive.
(ii) If M is an infinite ≤5-set-homogeneous 6-hypergraph whose automorphism group is not 3-primitive, then M is isomorphic to M 6 or its complement, or Aut(M ) preserves a separation relation on M .
As noted in Section 5.2, the hypergraphs M 3 , M 4 and M 6 belong in a family -they live on the same vertex set with Aut(M 3 ) < Aut(M 4 ) < Aut(M 6 ).We also note in Remark 3.3.3that N 3 is a two-graph -a 3-hypergraph such that any 4 vertices carry an even number of edges.The hypergraph N 3 is closely related to the set-homogeneous graph R(3) from [18] mentioned above.
The strategy for the construction of M 3 , N 3 , M 4 , M 6 is the same in each case, and analogous to the corresponding constructions in [18] (for graphs) and [25] (for digraphs).Let M be one of these k-hypergraphs, viewed as a structure in a language L with a single k-ary relation symbol.We first consider a suitable (in each case already known) homogeneous structure N with the same domain but different language L ′ , define M from N , and prove that Aut(M ) = Aut(N ) by showing they have the same ∅-definable relations (in the cases considered, the fact that Aut(N ) ≤ Aut(M ) ensures that M is not homogeneous.)We then consider an isomorphism σ : U → V between finite substructures of M , and show that the L ′ -structures induced on U and V from N are also isomorphic (though not necessarily via σ).It follows by homogeneity of N that there is g ∈ Aut(N ) with U g = V , and since Aut(M ) = Aut(N ) we find g ∈ Aut(M ), as required.The structures N which yield hypergraphs in this way are chosen from a small family of constructions already known to have interesting properties (see Section 2.1).It appears that other similar constructions do not yield set-homogeneous hypergraphs.
The results characterising these examples mostly use known results characterising permutation groups with a higher degree of homogeneity than transitivity -these are discussed in Section 2.2.The characterisation of M 3 uses a more direct bare-hands argument.
Lachlan and Tripp [32] classified finite homogeneous 3-hypergraphs, using the observation that their automorphism groups are 2-transitive, together with the classification of finite 2-transitive groups (so resting on the classification of finite simple groups).We also briefly consider finite set-homogeneous hypergraphs.We do not carry out a classification (though this looks fully feasible), but obtain the following result.Note that part (ii) shows the Enomoto argument mentioned above does not work for 3-hypergraphs.
(ii) There is a set-homogeneous 3-hypergraph on 7 vertices which is not homogeneous.
We also show that aspects of Lachlan's theory of finite homogeneous structures over an arbitrary finite relational language hold when homogeneity is weakened to set-homogeneity; see Theorem 5.1.3.
The paper is organised as follows.In the remainder of this introduction we briefly discuss the model theory of set-homogeneity.In Section 2 we give some preliminaries on certain specific treelike homogeneous structures and on some results on permutation groups, mainly of Cameron, that we use.We prove Theorem A in Section 3, and Theorems B and C in Section 4. We discuss finite set-homogeneous structures (in particular Theorem D) in Section 5.1, and consider some further directions and open questions in Section 5.2.

Model theory of set-homogeneity
We briefly discuss model-theoretic properties of set-homogeneity.Most background can be found in [11], and [37] can also be taken as a model-theoretic source.
A countably infinite first-order structure M is ω-categorical if it is determined up to isomorphism among countable structures by its first-order theory.By the Ryll-Nardzewski Theorem, a countably infinite structure M is ω-categorical if and only if its automorphism group is oligomorphic, that is, has finitely many orbits on M n for all n.It is immediate that any countably infinite set-homogeneous structure over a finite relational language has oligomorphic automorphism group, so is ω-categorical.
Recall that the age Age(M ) of a countably infinite relational structure M over a finite relational language L is the collection of finite structures which embed in M .Fraïssé's amalgamation theorem states that any two infinite homogeneous structures with the same age are isomorphic, and that a collection of finite L-strucures is the age of a homogeneous structure if and only if it is closed under isomorphism, substructure, joint embedding property, and has the 'amalgamation property'.It was noted in [18,Theorem 1.2] that we may in Fraïssé's Theorem replace 'homogeneous' by 'set-homogeneous' and 'amalgamation property' by 'twisted amalgamation property' (TAP), where a class C of finite structures has (TAP) if and only if for any A, B 1 , B 2 ∈ C and f i : It is well-known that an ω-categorical structure over a finite relational language is homogeneous if and only if its theory has quantifier-elimination.A natural weakening of quantifier-elimination is model-completeness: a theory T is model-complete if every formula is equivalent modulo T to an existential formula.It is shown in [21,Chapter 11,Section 8] with an attribution to Pouzet (see also [18, pp. 90-91]) that any infinite set-homogeneous structure M over a finite relational language is uniformly prehomogeneous; that is, for any finite A ≤ M there is finite B with A ≤ B ≤ M and with |B| bounded as a function of |A|, such that for any partial isomorphism f on M with domain A, if f extends to B then f extends to an automorphism of M .(Formally, the setting in [21] is for languages with a single relation symbol, but the extension to a finite relational language is routine).It is easily checked that any uniformly prehomogeneous structure is model-complete, and hence that any set-homogeneous L-structure is model-complete.

Notation
If G is a permutation group on a set X (sometimes written as (G, X)), we write x g for the image of x ∈ X under g ∈ G.For U ⊂ X and g ∈ G, let U g = {u g : u ∈ U }.For arbitrary functions we write the function to the left of the argument.
We view a k-hypergraph as a first order structure (M, E), where E is a kary relation which is assumed only to hold if all arguments are distinct, and to be invariant under permutations of the arguments (i.e. to be irreflexive and symmetric).If (M, E) is a k-hypergraph, we shall write x 1 . . .x k rather than {x 1 , . . ., x k } for an edge.A complete k-hypergraph is a k-hypergraph all of whose k-subsets are edges.The complement of the k-hypergraph (M, E), denoted (M, E) c (or just M c ), has the same vertex set M , but a k-subset of M is an edge of M c if and only if it is a non-edge of M .When we use the word 'subhypergraph' we always mean induced subhypergraph, i.e. the model-theoretic notion of substructure.
If < is a total order on X, and A, B ⊂ X, we write A < B to denote that ∀a ∈ A∀b ∈ B(a < b).
If (X, →) is a tournament (that is, a digraph such that for any distinct x, y ∈ X exactly one of x → y or y → x holds) and x ∈ X, then x + := {y ∈ X : x → y} and x − := {y ∈ X : y → x}.We denote by C 3 the tournament on {a, b, c} such that a → b → c → a.

Preliminaries
We review here some constructions of homogeneous structures and results on permutation groups which we use heavily.

Some homogeneous structures
As indicated in the Introduction, our strategy for finding set-homogeneous hypergraphs (M, E) is to find certain other very specific homogeneous structures on the same domain and with the same automorphism group.We here give a brief review of the structures used.
First, we recall the linear betweenness relation, circular order, and separation relation which are derivable from a linear order.If (X, ≤) is a linear order, then a linear betweenness relation B(x; y, z) can be defined on X, putting A circular ordering K(x, y, z) is definable on X with the rule Given a circular order K on X, a separation relation on X is defined by S(x; y; z, w) ⇔ K(x, y, z) ∧ K(x, w, y) ∨ K(x, z, y) ∧ K(x, y, w) .
Here, if x, y, z, w are distinct, then S(x, y; z, w) says that z, w lie in distinct segments with respect to x, y of the circular order (and vice versa).Axioms for these can be found in [1, Part I], and it can be shown that any structure satisfying these axioms arises from a linear order in this way.
Next, we introduce C-relations.Following Section 10 of [1], a C-relation is a ternary relation C on a set M satisfying (C1)-(C4) of the following axioms (with free variables all universally quantified -we omit these quantifiers); it is proper if it also satisfies (C5) and (C6).It is shown in [1,Theorem 11.2] that if (X, ) is a lower semilinearly ordered set (a partial order such that for each a the set {x : x a} is totally ordered, and such that any two elements have a common lower bound), then there is a natural C-relation on the set S of maximal chains (totally ordered subsets) of X; here C(x; y, z) holds if and only if x ∩ y ⊆ y ∩ z (where the chains x, y, z are viewed as subsets of X).Furthermore [1, Theorem 12.4] any C-set (M, C) arises in this way, with M a 'dense' set of maximal chains of some lower semilinear order (X, ) -the density here means that any node a ∈ X will lie in some chain of M .The semilinear order (X, ) is canonically constructed from (M, C) -indeed it is first-order interpretable without parameters in (M, C), as a quotient of an equivalence relation on M 2 ; we shall view (M, C) as coming from such (X, ).Configurations for a C-relation (and D-relation below) are shown in Figure 1.With (X, ) and (M, C) as above, for a ∈ X, let S a be the set of chains in M which contain a.There is a natural equivalence relation E a on the set S a : we put E a xy if and only if there is b ∈ X with a ≺ b such that x, y both contain b.We call the E a -classes cones of (M, C), and say that (M, C) is regular if the number of cones at a does not depend on a, and is k-regular if this number is k; in this case, following [1, Section 10], the branching number of (M, C) is k + 1.It is well-known (see for example [10, pp. 159, 161], or [1,Theorem 12.6]) that for each k ∈ N ≥2 ∪ {∞} there is up to isomorphism a unique countably infinite dense k-regular proper C-set, and this structure is homogeneous.
Next, we briefly introduce D-relations, as axiomatised in Part V of [1].An We tend to think of a D-relation as holding on the set of 'directions' of a general betweenness relation (as defined in Part V of [1]); in fact, by [1,Theorem 26.4] any D-relation arises in essentially this way.If (M, E) is a graph-theoretic unrooted tree whose vertices have degree at least three, then there is a Drelation (not satisfying (D6)) on the set of ends; for distinct ends x, ŷ, ẑ, ŵ we put D(x, ŷ; ẑ, ŵ) if and only if there are x ∈ x, y ∈ ŷ, z ∈ ẑ and w ∈ ŵ such that x∪y and z ∪w are vertex-disjoint two-way infinite paths.Observe that if (M, D) is a D-set and a ∈ M , then there is an induced C-relation C a on M \ {a} with 'downwards direction a' -define C a (x; y, z) to hold if and only if D(a, x; y, z) holds.We shall say that the there is a unique countably infinite dense proper k-branching D-set, and this structure is homogeneous (existence and uniqueness follow from [1, Theorems 12.6 and 22.1], and homogeneity from the 'First Variation' in [10,

p.159]).
There is a natural notion of a C-relation C on M being compatible with a total order on M -this is introduced and exploited in Section 2.1.There is likewise a notion of a D-relation being compatible with a circular order, but this appears not to lead to further set-homogeneous hypergraphs.
We mention one further construction -the dense local order of [8], which is one of the three countably infinite homogeneous tournaments classified by Lachlan in [31].Following [8], a local order is a tournament such that all outneighbourhoods x + and in-neighbourhoods x − are totally ordered by →.The dense local order is easiest to describe as the unique countably infinite tournament T = (Z, →) obtained by distributing a countably infinite set Z of points densely on the unit circle, no two antipodal, and putting x → y if the clockwise distance from x to y on the circle is less than the anticlockwise distance.It is the countable homogeneous tournament obtained as the Fraïssé limit of the collection of all finite local orders; by [8, Theorem 6.2] it is the unique (up to isomorphism) countable tournament on at least 4 vertices containing C 3 and such that each set (x + , →) and (x − , →) is a dense linear order without endpoints.As noted by Cameron in [8, pp. 57-58] this tournament is isomorphic to its complement, and if γ is such an isomorphism, then the group H := Aut(T ), γ has Aut(T ) as a subgroup of index 2.In particular, H is 2-transitive, not 2-primitive, and preserves a separation relation on Z.
These examples have been heavily explored in the literature, from various points of view.For example, they are among the few known examples of countably infinite structures M whose automorphism groups Aut(M ) are oligomorphic, primitive, and have the property that if f (k) denotes the number of orbits of Aut(M ) on k-element subsets of M , then f (k) is bounded above exponentially.This viewpoint is developed in [10].Among homogeneous structures, these structures each have the rare property that their age is well-ordered under embeddability.For recent work in this direction see for example Conjecture 1 of [5] that for a homogeneous ω-categorical relational structure these conditions are co-incident, and coincide also with 'monadic NIP'.The automorphism groups of homogeneous C and D-sets are Jordan groups -this was a motivating theme of [1].
Many of our arguments with C and D-relations and local orders are very pictorial.We include some diagrams to indicate the intended configurations, but encourage the reader to draw others.

Homogeneity in permutation groups
Our results on set-homogeneity use and mimic several earlier results on homogeneity of permutation groups.First, recall the following result of Cameron which lies in the background.
Theorem 2.2.1.[7] Let (G, X) be an infinite permutation group which is khomogeneous for all k but not k-transitive for some k.Then G preserves on X a linear order, circular order, linear betweenness relation, or separation relation.
We use below a number of refinements of this result, listed together below.
Theorem 2.2.2.Let G be a permutation group on an infinite set X. Then the following hold.
(iii) (Cameron [7,Theorem 5.1]) Suppose that G is 4-homogeneous but not 3-transitive.Then G preserves a circular order or linear betweenness relation on X.
(iv) (Cameron [9,Remark p. 245] -see [33,Proposition 1.3] for proof ) If G is 5-homogeneous, 3-transitive but not 3-primitive, then either G preserves a separation relation on X or G preserves on X a structure elementarily equivalent to the universal homogeneous 3-branching D-set.

Countably infinite set-homogeneous 3-hypergraphs
In this section we first describe an example of a countably infinite set-homogeneous 3-hypergraph which is not 2-transitive (Subsection 3.1), thereby proving one direction of Theorem A(i).We prove the other direction in Section 3.2, and part (ii) in Section 3.3.

Construction of an example
In this subsection we construct from a totally ordered C-set a countably infinite set-homogeneous 3-hypergraph (M, E) whose automorphism group G is not 2transitive.The ordered C-set has appeared previously, in various guises -see for example the structure ∂P T 3 on p.162 of [10], or the structure M 2 in [27, Section 3], or (L, C, ≺) of [4,Proposition 3.14].Suppose that on the set M there is defined a total order ≤ and a C-relation C. We say that the relations C and ≤ are compatible, and that (M, C, ≤) is a (C, ≤)-set if all cones and all sets S a (see Section 2.1) are convex with respect to ≤.However, we only consider the notion when (M, C) is a 2-regular C-set, and under this condition compatibility is equivalent to the condition that whenever x < y < z we have C(x; y, z) ∨ C(z; x, y) (cf.[4, Section 3.5]); we use the definition of compatibility in this latter form.Informally, if we draw a lower semilinear order (X, ) in the plane in the natural way, with maximal chains never 'crossing', then the natural left-to-right total order on the set of maximal chains is compatible with the C-relation defined above.We shall say that the One configuration witnessing (C8) is shown in Figure 2. Note that (C8) implies the density condition (C7).The following result has been long known and attribution is difficult.Proof.We omit the details.For existence, the existence of the corresponding semilinear order is well-known (it is the countable '2-homogeneous tree' of 'positive type' and 'ramification order 2' from [16]).We may take any countable dense set of maximal chains from this semilinear order, with the natural induced (C, ≤)-set structure.The structure so obtained is denoted ∂P T 3 in [10, p. 162].Uniqueness can be proved by a routine back-and-forth argument.This is done explicitly in [27,Theorem 4.6], though some translation to the language of that paper is necessary.See also [4,Proposition 3.14] (where, as discussed with one of the authors, the condition 'strongly dense' was inadvertently omitted).
For the rest of this section, (M, C, ≤) denotes the structure identified in Theorem 3.1.1.
We now define the edge relation E to hold of a triple xyz of distinct elements of M if (Here x < {y, z} means x < y ∧ x < z.) Thus in Figure 2 xyz is an edge but w 2 yz is not.Let M 3 be the hypergraph (M, E), and put G = Aut(M 3 ).
Proof.For convenience, observe that the pictures in Figure 3 correspond to 4-sets carrying 0,1,2,3,4 hypergraph edges respectively.First, we observe that the ordering < on M is ∅-definable in (M, E) and hence G-invariant, since for all distinct y, z ∈ M we have y < z if and only if there are distinct u, v ∈ M \ {y, z} such that the only edges on {u, v, y, z} are uvz and yvz.Also (though we do not need this), the relation C is definable from E: we have C(x; y, z) if and only if We next show that M 3 is set-homogeneous.We prove by induction on |U | that if U , V are isomorphic finite substructures of (M, E) then they are isomorphic as (C, ≤)-sets (possibly with a different isomorphism) and hence by homogeneity of (M, C, ≤) there is g ∈ G with U g = V .This is immediate by inspection if |U | ≤ 3.
So suppose that U, V are isomorphic finite substructures of (M, E) with |U | = |V | ≥ 4, and let σ : U → V be an isomorphism.Define Pictorially, S and T are as in Suppose first that S = ∅.Let S = {a 1 , . . ., a r } with a 1 < . . .< a r .Then a i < b for any i ∈ {1, . . ., r} and b ∈ U \ S and C(a i ; a j , a k ) whenever i < j < k ≤ r, and C(a i ; a j , b) whenever i < j ≤ r and b ∈ U \ S; also C(a i ; b, c) for any i ∈ {1, . . ., r} and b, c ∈ U \ S. The corresponding assertions hold for S ′ and V .Since σ(S) = S ′ we have σ(U \ S) = V \ S ′ .Thus U \ S and V \ S ′ carry isomorphic hypergraphs of size smaller than |U |, so by induction carry isomorphic (C, ≤)-sets.It follows that U and V carry isomorphic (C, ≤)-sets, so by homogeneity of (M, C, ≤) there is h ∈ Aut(M, C, ≤) with U h = V .Such h also preserves E (since E is definable in terms of C, ≤), so h ∈ G.
Thus, we may suppose S = ∅, and similarly T = ∅.There is a partition of U into two parts P, Q, with the following property: (*) |P | ≥ 2, |Q| ≥ 2, and for all p ∈ P and distinct q, q ′ ∈ Q we have E(p, q, q ′ ), and for all q ∈ Q and distinct p, p ′ ∈ P we have ¬E(p, p ′ , q).Indeed, in the underlying tree structure induced from (M, C, ≤), P will be the left hand cone at the root, and Q the right hand cone.In particular, P < Q.
We claim that this is the unique two-part partition of U satisfying (*).Indeed, suppose that U = P ′ ∪ Q ′ is another such partition, and that P ′ contains p, p ′ where p lies in the left hand cone P and p ′ in the right hand cone Q.Now Q ⊆ P ′ ; for if q ∈ Q∩Q ′ then E(p, q, p ′ ) contradicting that q ∈ Q ′ and p, p ′ ∈ P ′ .Thus, if p ′′ ∈ Q \ {p ′ } and q ∈ Q ′ then p ′′ ∈ P ′ and q ∈ P , so E(q, p ′ , p ′′ ) by (*) applied to (P, Q).This however contradicts (*) for P ′ , Q ′ .
It follows that σ maps P to the left hand cone P * of V and Q to the right hand cone Q * of V .In particular the hypergraphs induced on P and P * are isomorphic, as are those on Q and Q * .Since |P | < |U | and |Q| < |U |, by induction the structures induced on P and P * are isomorphic as (C, ≤)-sets, as are those on Q and Q * .It follows that the (C, ≤)-structures induced on U and V are isomorphic, so again by homogeneity of (M, C, ≤) there is h ∈ Aut(M, C, ≤) with U h = V , and such h lies in G.

Classification in the not-2-transitive case
The goal of this subsection is to show that M 3 is the unique countable sethomogeneous 3-hypergraph whose automorphism group is not 2-transitive.The idea is to recover from E the relations C and ≤, show that (M, C, ≤) is a strongly dense 2-regular (C, ≤)-set, and apply Theorem 3.1.1.The first three lemmas below hold for finite as well as countably infinite structures.First, we record the following easy lemma, used throughout the paper.
Proof.Suppose that U and V each have i > 0 edges.Each edge of U omits one element of U , so the intersection of the edges of U is a subset S U of size k + 1 − i, and similarly the intersection S V of the edges of V has size k + 1 − i.It follows that any bijection σ : U → V with σ(S U ) = S V gives an isomorphism (U, E) → (V, E), and by (k + 1)-set-homogeneity some such σ lifts to g ∈ Aut(N, E).
Observe that the arc 2 → 3 lies on at least two copies of C 3 , namely 234 and 235.It follows by 2-homogeneity of Aut(T, →) on T that every arc lies on at least two copies of C 3 .
Proof.By set-homogeneity, K is 2-homogeneous on N .Since K is not 2transitive it preserves a tournament relation → on N ; here the set of pairs (a, b) with a → b is one of the two K-orbits on ordered pairs of distinct elements.We assume → is not a total order, so (N, →) embeds a copy of C 3 .Now |N | ≥ 4, since otherwise (N, →) ∼ = C 3 and Aut(N, E) = S 3 which is 2-transitive.Thus, (N, →) embeds paths of length 3 as well as C 3 , and we may suppose that the copies of C 3 are the edges of (N, E).By transitivity of K on N , if for some a ∈ X we have that a + (respectively a − ) embeds a copy of C 3 , then this holds for all x ∈ N .
If neither a + nor a − embeds a copy of C 3 , then (N, →) is a local order (see Section 2.1).Easily, if |a + | = 1 for all a then |a − | = 1 for all a and we have (N, →) ∼ = C 3 , a contradiction as above.Thus, using rigidity of finite total orders we may assume that each a + and a − is infinite.By 3-set-homogeneity each set a + and a − is densely ordered, so by [8, Theorem 6.2] (N, →) is isomorphic to the dense local order (S, →) (one of the three homogeneous tournaments classified by Lachlan in [31]).Now as noted in Section 2.1, there is a permutation γ of N (of order 2) which induces an isomorphism from (N, →) to its reverse.Now γ preserves E, so γ ∈ K, contradicting that → is K-invariant.
Thus, we may suppose that each x + embeds C 3 .It follows by Lemma 3.2.2 that each x − also embeds C 3 .This however means that (N, →) contains two non-isomorphic 4-vertex tournaments each giving a hypergraph with exactly one edge (a copy of C 3 dominated by a vertex, and a copy of C 3 dominating a vertex).This is impossible by Lemma 3.2.1.
We aim to show that any countably infinite set-homogeneous but not 2homogeneous 3-hypergraph (N, E) is isomorphic to M 3 .Consider such (N, E), and let K = Aut(N, E).By Proposition 3.2.3,K preserves a linear order ≤ on N , and by 2-homogeneity, this order is dense without endpoints.Our goal is to reconstruct the C-relation on N .
Let U = {x, y, z, w} be a 4-vertex substructure of the structure (M, E, ≤) from Section 3.1, with x < y < z < w.It is easily checked (compare Figure 3) that if U has one edge then this is xyz, if U has 2 edges then these are xzw and yzw, and if U has 3 edges then these are all except yzw.Also, all these cases are realised.We first aim to recover this behaviour on (N, E) (of course, up to reversal of the ordering).Note that in N , by Lemma 3.2.1 any two 4-vertex substructures with the same number of edges are in the same K-orbit, so the edges must be distributed in the same way with respect to the order, that is, the structures are isomorphic as (E, ≤)-structures.
We observe first that (N, E) contains both edges and non-edges, since otherwise K = Sym(N ) and so is 2-transitive.We shall amalgamate edges and non-edges in all possible ways over 2-sets.Since we use similar arguments for 4-hypergraphs in the next section, and the arguments may be applicable for k-hypergraphs for larger k, we set them up in greater generality in the next two lemmas.Lemma 3.2.4.Let k ≥ 4 and let (H, E) be a ≤(k + 1)-set-homogeneous countably infinite k-hypergraph whose automorphism group preserves a total order < on H.
Proof.(i) First choose a non-edge w 1 . . .w k of H with w 1 < . . .< w k .Suppose first i < j.There is v such that w j−1 < v < w j and {v, w 1 , . . ., w k } \ {w i } is an edge.In the case when j < i, choose v with w j−1 < v < w j such that {v, w 1 , . . ., w k } \ {w i−1 } is an edge.In both cases put u m = w m for m < j, u j = v, and u m = w m−1 for m > j to obtain the required set.
The conclusion of Lemma 3.2.4 is that the formula S ij is realised in (H, E) for all i, j ∈ {1, . . ., k + 1} with |i − j| ≥ 2 and T i is realised for each i = 1, . . ., k.
If a (k+1)-set has i edges, then these edges intersect in k+1−i elements, and under an assumption of (k + 1)-set-homogeneity of (H, E) and the invariance of <, these k + 1 − i elements are determined by i.For each i = 1, . . ., k we say that H satisfies P i J where J is a (k + 1 − i)-subset of {1, . . ., k + 1}, if H has a (k + 1)-set {u 1 , . . ., u k+1 } (with u 1 < . . .< u k+1 ) with exactly i hypergraph edges, which intersect in the elements indexed by J. (In the case where i = k and so |J| = 1, we write P k j rather than P k {j} .)We say H satisfies P i * if H has no (k + 1)-set with i edges.Thus, for each i either P i J holds for exactly one (k + 1 − i)-subset J of {1, . . ., k + 1}, or it holds for no J and P i * holds.Furthermore, by Ramsey's theorem either there is a (k + 1)-vertex set with k + 1 edges, or there is a (k + 1)-vertex set with no edges (possibly both).Lemma 3.2.5.Assume the conditions of Lemma 3.2.4.Then (i) P k J ensures that all formulas S ij are realised where i ∈ J and j ∈ J, and (ii) for i = 1, . . ., k, a (k + 1)-tuple satisfying P k J realises T i if and only if just one of i, i + 1 lies in J.
Proof.If P k J holds, then any u 1 < . . .< u k+1 carrying k edges has the edge omitting u i for each i ∈ {1, . . ., k + 1} \ J, and has the non-edge omitting u j for each j ∈ J.Both parts follow.
We now revert to our context where (N, E) is a ≤4-set-homogeneous 3hypergraph whose automorphism group K preserves a total order ≤ on N .For each i = 1, 2, 3 either P i J holds for some J ⊂ {1, .By examining the possibilities, we easily see that the first case gives (i) and the second case gives (ii).
We shall assume that Case (ii) holds; this is justified since (i) is obtained from (ii) by reversing the order (in fact, if (i) holds, we obtain the complement of M 3 , which as a hypergraph is isomorphic to M 3 ).Lemma 3.2.7.For each i ∈ N with 0 ≤ i ≤ 4 there is a 4-vertex substructure of (N, E) with exactly i edges.
Proof.For i = 1, 2, 3 this follows from our Case (ii) assumption.For the case i = 0, fix a non-edge uvw of with u < v < w.There is a < u with auv a nonedge, and by inspecting the possibilities in our Case (ii) assumption, {a, u, v, w} has 0 edges.For the case i = 4 consider an edge pqr with p < q < r, and some d < p with dpq an edge -the set {d, p, q, r} must have four edges.
Next, we define a ternary relation C on N as follows.For x, y, z ∈ N , we put C(x; y, z) if and only if one of the following holds.Proof.Axioms (C1) and (C4) of Section 2.1 follow immediately from the above definition.(C2) is immediate if y = z ∧ x = y, so suppose that C(x; y, z) holds and x, y, z are distinct.Either x < Min{y, z} and Exyz, or Max{y, z} < x ∧ ¬Exyz.In the first case, we do not have y < Min{x, z} so C(y; x, z) cannot hold through (b), and as Exyz holds, C(y; x, z) cannot hold through (c).The argument is similar in the second case.Thus (C2) holds.
To prove (C3), suppose C(x; y, z) holds, and w ∈ N .In case (a), where y = z = x, either w = y = z and C(x; w, z) holds, or w = y and C(w; y, z) holds, as required.
Suppose case (b) holds.If w = x then C(w; y, z) holds, if w = y then C(x; w, z) holds, and if w = z then C(x; w, z) via (a).Thus, we may suppose x, y, z, w are distinct.Now as Exyz holds, {x, y, z, w} has 1,2,3 or 4 edges.If there is just one edge, then P 1 {1,2,3} (from Lemma 3.2.6(ii))yields that Max{x, y, z} < w and yzw is a non-edge, so C(w; y, z) by (c).If there are 2 edges then as P 2 {3,4} holds and as C(x; y, z) arises from (b), we must have that w < Min{y, z} and Ewyz, so C(w; y, z) holds by (b).If there are 3 edges then by P 3 {1} as Exyz we must have x < w.Since all the triples of {x, y, z, w} with element x are edges, we must have Exwz, so C(x; w, z) holds by (b).Finally, if there are 4 edges, then if w < Min{y, z} then C(w; y, z) holds, and otherwise C(x; w, z) holds, in each case by (b).
The proof when C(x; y, z) holds through case (c) is similar.Thus, we have established (C3).
For axiom (C5), if y = z then any x = y satisfies C(x; y, z).And if y = z then using 2-homogeneity of K and the existence of edges we find x < Min{y, z} with Exyz, and then C(x; y, z) holds.
Finally, for (C6), suppose x = y.Suppose first x < y.Again using 2homogeneity and existence of edges, there is z > y with Exyz, and then y = z ∧ C(x; y, z) holds.Similarly, if y < x, there is z < y with ¬Exyz, and again y = z ∧ C(x; y, z), the latter by (c).
For 2-regularity, suppose for a contradiction that there are distinct x, y, z such that ¬C(x; y, z) ∧ ¬C(y; x, z) ∧ ¬C(z; x, y).We may suppose x < y < z.Then xyz is a non-edge by (b) as ¬C(x; y, z), but xyz is an edge by (c) as ¬C(z; x, y), a contradiction.Lemma 3.2.9.The order ≤ on N is compatible with C, and (N, C, ≤) is a strongly dense (C, ≤)-set.
Proof.In this 2-regular context, for compatibility we must show that if x < y < z then C(x; y, z) ∨ C(z; x, y).This is immediate -if xyz is an edge then C(x; y, z) holds, and if xyz is a non-edge then C(z; x, y) holds.
It remains to prove strong density, i.e. that (C8) holds.So suppose that C(x; y, z) holds.Again, we may suppose x < Min{y, z}, the case when Max{y, z} < x being handled similarly.
We suppose first y = z.Then there is w 1 with x < w 1 < y and xw 1 y an edge, and for such w 1 we have C(w 1 ; y, y) ∧ C(x; y, w 1 ), and likewise there is w 2 > y with xyw 2 an edge, again yielding C(w 2 ; y, y) ∧ C(x; y, w 2 ).
Suppose now y = z.We may suppose y < z.By Lemma 3.2.7 there are p < q < r < s in N such that {p, q, r, s} has 4 edges.Since xyz is an edge, by 3-set-homogeneity of (N, E) we may choose g ∈ K with {p, r, s} g = {x, y, z} and put w 1 = q g .By K-invariance of < we have x < w 1 < y and xyw 1 and w 1 yz are edges, so C(w 1 ; y, z) ∧ C(x; y, w 1 ) holds.Likewise there is w 2 > z such that {x, y, z, w 2 } has three edges, yielding again C(w 2 ; y, z) ∧ C(x; y, w 2 ).

Proof of Theorem A(i).
The existence assertion follows from Proposition 3.1.2.For the uniqueness characterisation, suppose that (N, E) is a ≤4-set-homogeneous countably infinite 3-hypergraph whose automorphism group K is not 2-transitive.By Lemma 3.2.3 and Lemmas 3.2.8 and 3.2.9,there are a K-invariant total order ≤ on N and compatible K-invariant 2-regular C-relation C on N so that (N, C, ≤) is strongly dense.It follows from Theorem 3.1.1that (N, C, ≤) ∼ = (M, C, ≤).By our assumption that Case (ii) of Lemma 3.2.6 holds, it follows that E is defined from C and ≤ in N in the same way as in M , and hence that (N, E) ∼ = M 3 .If instead we had assumed that any 4-vertex set in N with two edges has the edges intersecting in the first two elements, then (N, E) would be isomorphic to the hypergraph complement M c 3 of M 3 .However, since We consider a further set-homogeneous 3-hypergraph associated with the countably homogeneous local order T = (Z, →) described in Section 2.1.Let H be the group described there of automorphisms and anti-automorphisms of T , which has Aut(T ) as a subgroup of index 2. Our arguments below heavily appeal to the description of (Z, →) as consisting of points on the unit circle (see Section 2.1).We frequently use that in this representation of T , Aut(T ) preserves the natural circular ordering on Z, and H preserves the induced separation relation S; this is easily verified.There is a natural notion of a subset U of Z being convex with respect to the circular order; this means that for any distinct x, y ∈ U and distinct z, w ∈ Z \ U , ¬S(x, y; z, w) holds.We define a ternary relation R on Z, putting R(x; y, z) if and only if Clearly H ≤ Aut(Z, R).When discussing (Z, R) (and (Z, E) below) we often refer to the underlying tournament from which they are defined -even though these structures do not determine the tournament relation →.
The lemma below is useful to us and may have independent interest: the group H is significant, since among primitive oligomorphic groups which are not k-homogeneous for all k, H has the slowest known growth for the function f (k) mentioned in Section 2.1.
Using that H is transitive on Z and on {(x, y, z) : R(x; y, z) holds}, there is h ∈ H and a, b, c ∈ Z with R(b; a, c) such that gh fixes a, b, c.We may suppose (adjusting h if necessary) that b → a.Now for any u ∈ Z \ {a, b, c} we have b → u ⇔ R(a; b, u) ∨ R(u; b, a) , and for distinct such u, u ′ we have u → u ′ ⇔ R(u; b, u ′ ).Also if b → u and u ′ → b, then u → u ′ ⇔ ¬R(b; u, u ′ ).Orientations of pairs u, u ′ with u → b and u ′ → b and pairs involving a, c are likewise easily recoverable.Thus, as gh ∈ Aut(Z, R), also gh ∈ Aut(Z, →) < H, so g ∈ H.
We shall say finite X ⊂ Z is linear if there is an enumeration X = {x 1 , . . ., x t } such that R(x j ; x i , x k ) holds whenever i < j < k or k < j < i.We call Then R(a; c i , d Let L R be the language with just the relation symbol R. Define a 3hypergraph N 3 = (Z, E) whose edges are the 3-sets of (Z, R) which satisfy R under some ordering, that is, lie in a segment making an angle less than π at the centre.Proposition 3.3.2.The hypergraph N 3 is set-homogeneous, but not 3-homogeneous.In particular its automorphism group is not 2-primitive, and does not induce the full symmetric group on triples satisfying E.
Proof.We first show that G = Aut(N 3 ) preserves R, and hence equals H. Indeed, if xyz is an edge, then we have R(y; x, z) ↔ [∃u∃v(uxy, vyz are edges and uxz, vxz are non-edges)].
In particular, as R(x; y, z) → ¬R(y; x, z), (Z, E) is not 3-homogeneous.Also, G is not 2-primitive, since G x preserves an equivalence relation on Z \ {x} with classes {y : x → y} and {y : y → x}; indeed, distinct y, z are in the same equivalence class if and only if R(y; x, z) ∨ R(z; x, y) holds.
Next, we show that N 3 is set-homogeneous.For any finite U ⊂ Z, define It is easily seen that ∼ U is an equivalence relation on U , and its classes are convex in the (clockwise) circular order induced from N 3 and are complete subhypergraphs.We say that U is balanced if all ∼ U -classes have size 1.
Claim 1. Suppose that finite U ⊂ Z is balanced.Then |U | is odd and Aut(U, E) = Aut(U, R) and equals the dihedral group D |U| .
Proof of Claim.For any distinct x, y ∈ U with x ∼ U y and x → y, there is z ∈ U such that x and y lie in opposite segments with respect to z, that is, we have y → z → x.Furthermore, assuming further there is no w ∈ U with x → w → y, such z is unique; for if z, z ′ ∈ y + ∩ x − then z ∼ U z ′ .Also, if z ∈ U and z + ∩ U = {x 1 , . . ., x r } with z = x 0 → x 1 → . . .→ x r , then for each i = 0, . . ., r − 1 there is unique y i ∈ z − ∩ U with x i+1 → y i → x i , and U = {x 0 , x 1 , . . ., x r , y 1 , . . ., y r }, so |U | = 2r + 1.
It is now easily seen that with D 2r+1 acting in the natural way on U preserving the induced separation relation S, we have To see that Aut(U, E) ≤ D 2r+1 , suppose g ∈ Aut(U, E) fixes x 0 , x i with i > 0 (in the notation above).We claim that g fixes x 1 , so suppose i > 1.Indeed, g fixes setwise {x j : 0 < j < i} = {w ∈ U : (∀y ∈ U )(Eyx 0 x i → Eyx 0 w)}.Thus g fixes y 1 , which is the unique element of U \ {x 0 , . . ., x i } such that there is exactly one w ∈ {x j : 0 < j ≤ i} (namely x 1 ) with ¬Ey 1 x 0 w.Hence g fixes x 1 .Now y 1 ∈ U is unique in U such that x 0 x 1 y 1 is a non-edge, so g fixes y 1 .Now x 2 is unique in U \ {x 0 , x 1 } such that |{y ∈ U : x 1 x 2 y is a non-edge}| = 1, so g fixes x 2 .Continuing this way, we find g = 1, yielding Aut(U, E) = D 2r+1 and hence the claim.Now suppose that σ : U → V is an isomorphism between finite subhypergraphs of N 3 .Since ∼ U is defined from E, σ maps ∼ U -classes to ∼ V -classes.Let U 1 , . . ., U n be the ∼ U -classes of U listed in the clockwise cyclic ordering, and for each i let a i ∈ U i , let b i = σ(a i ), let V i be the ∼ V -class of b i , and put A = {a 1 , . . ., a n }, and B = {b 1 , . . ., b n }.Then A and B are balanced, and by the claim σ induces an isomorphism (A, R) → (B, R).We shall say that σ is positive if it preserves the positive circular orientation on A, and negative otherwise (one of these holds, by the claim).
Since σ maps ∼ U -classes to ∼ V -classes, it follows that for each i = 1, . . ., n the equivalence classes U i and V i have the same size t i .For each i = 1, . . ., n write U i = {a i1 , . . ., a iti } with a ij → a ik whenever j < k.For each i = 1, . . ., n we also put Finally, define α : U → V by putting σ(a ij ) = b ij for each i = 1, . . ., n and j = 1, . . ., t i .It suffices to prove the following claim.
Claim 2. The map α : (U, R) → (V, R) is an isomorphism.Proof of Claim.For triples within a ∼ U -class, σ preserves R since it preserves or reverses the ordering given by →.For triples meeting three distinct ∼ Uclasses, α preserves R as σ| A does, and as elements outside a ∼-class are →related to all elements of the ∼ U -class in the same way.For triples containing two elements a ik , a il (k < l) from one class U i , and one element a jm from another class U j , suppose first that σ is positive, and that a i → a j .Then a ik , a il ∈ a − jm , and a ik → a il , so R(a il ; a ik , a jm ).We have b ik → b il and b ik , b il ∈ b − jm , so R(b il ; b ik , b jm ) as required.The other cases (where a j → a i , and where σ is negative) are similar.
Given Claim 2, it follows by homogeneity of (Z, R) (see Lemma 3.3.1)that α is induced by some g ∈ Aut(Z, R) = Aut(Z, E), and we have U g = V as required.

Proof of Theorem A(ii). See Proposition 3.3.2. ✷
Remark 3.3.3.The example N 3 is a two-graph, namely a 3-hypergraph with the property that any four vertices carry an even number of hypergraph edges (in this case, 2 or 4).This notion was introduced by D.G. Higman -see [20] or [12] for background (including the infinite case).
In this section we first apply methods from Section 3, in particular Lemma 3.2.4 and 3.2.5, to prove Theorem B(i).Then in Section 4.2 we prove Theorem B(ii) and (iii), basing our construction on the (unordered) 2-regular countable dense proper C-set.Theorem C is proved in Section 4.3, exploiting a 3-branching D-set.

Proof of Theorem B(i)
First, observe that if (M, E) is an infinite set-homogeneous k-hypergraph with k ≥ 4 whose automorphism group G is not 2-transitive, then G is 3-homogeneous, so by Theorem 2.2.2(i),G preserves a linear order < on M .We now consider the case where k = 4.We have not tried hard to apply the methods for larger k.
Proof of Theorem B(i).Let (M, E) be a ≤5-set-homogeneous countably infinite 4-hypergraph whose automorphism group G is not 2-transitive.By Theorem 2.2.2(i) there is a G-invariant dense total order < on M , and we adopt the notation S ij and P k J from Section 2.2.By Lemmas 3.2.4 and 3.2.5, each formula S ij with |i − j| ≥ 2 is realised by a 5-element substructure of M , and for each i = 1, 2, 3, 4, M contains a 5-tuple realising T i .There are 12 such S ij , namely, S 13 , S 14 , S 15 , S 24 , S 25 , S 35 (where i < j) and the corresponding formulas with i > j namely S 31 , S 41 , S 51 , S 42 , S 52 , S 53 .By Lemma 3.2.5, the ways in which conditions P k J ensure that the formulas S ij and T i are realised are determined by Table 2.
By Lemma 3.2.5, we must show that there do not exist sets J 1 , J 2 , J 3 , J 4 so that if P i Ji hold for each i = 1, . . ., 4 then all formulas S ij and T 1 , T 2 , T 3 , T 4 are realised.Since |J 1 | = 4 and |J 2 | = 3 and they are both subsets of {1, 2, 3, 4, 5}, |J 1 ∩ J 2 | ≥ 2. We consider all possible 2-sets which could lie in the intersection, using symmetry (essentially, reversing the order) to reduce the number of cases.As a small abuse, we shall write ijK for the set J = {i, j} ∪ K.
1. M realises P 1 12J and P 2 12K .These do not ensure realisation of T 1 , S 13 , S 14 , S 15 , S 24 and S 25 (see Table 2).So the latter must arise from P3 J and P 4 J .The only cases which realise S 13 are the following cases.
(a) P 4  3 which needs T 1 , S 14 , S 15 from some P   .This is almost the same as the last case.These conditions do not realize S 13 , S 14 , S 41 , S 15 , S 42 which need to be realized by P 3 and P 4 .As before, the only possibilities of S 13 are: (a) P 4  3 which needs S 41 , S 14 from some P Proof.As noted in Section 2.1, (M, C) is a homogeneous structure.The 4-sets prescribed to form edges of M 4 are those as in Figure 6.

Figure 6
Let G = Aut(M 4 ).We observe first that G preserves the relation C. Indeed, C is ∅-definable in (M, E): for x, y, z ∈ M we have that C(x; y, z) holds if and only if y = z ∧ x = y ∨ ∃u∃v(x, y, z, u, v are distinct and has only the edge yuvz) .
See Figure 7  We show next that M 4 is set-homogeneous.So suppose U, V are finite subsets of M which carry isomorphic induced 4-hypergraphs, with σ : (U, E) → (V, E) an isomorphism.We show by induction on |U | that (U, C) ∼ = (V, C), from which it follows by homogeneity of (M, C) that there is g ∈ Aut(M, C) = The following result now gives us Theorem B(iii).Proof.First, we observe that the D-relation can be defined from the hypergraph structure.Given distinct x, y, z, w ∈ N , put D(x, y; z, w) if and only if there is u ∈ N \ {x, y, z, w} such that {x, y, z, w, u} has only the non-edges xyuz, xyuw.It can be checked that this correctly recovers the D-relation from E, so Aut(N 4 ) = Aut(N, D).
Homogeneity of (N, D) and inspection of possible 5-element substructures easily yields that N 4 is ≤5-set-homogeneous.It is not 6-set-homogeneous, as N 4 has 6-vertex complete subhypergraphs corresponding to the non-isomorphic D-sets in Figure 9.

Set-homogeneous 6-hypergraphs
In this section we prove Theorem C. The main point is the existence assertion (i), restated below.The example is defined from a D-set.Proposition 4.3.1.There is a countably infinite set-homogeneous 6-hypergraph M 6 = (M, E) whose automorphism group G is 3-transitive but not 3-primitive.Proof.Our starting point is the group J described in Theorem 5.1 of [9].The language in that paper is different, but J is the automorphism group of the unique countable homogeneous D-set (M, D) with branching number 3, as described in Section 2.1 above and in Section 32 of [1].It is easily checked and noted in [9, Theorem 5.1]) that J is 5-homogeneous but has two orbits on the collection of subsets of size 6.We define a 6-hypergraph M 6 on M in which the edges are sets whose induced D-structure has the isomorphism type in Figure 10.Let E denote the resulting 6-ary edge-relation on M .Clearly J = Aut(M, D) ≤ Aut(M 6 ).
Claim 1. Aut(M 6 ) = J.Proof of Claim.We must define D from E. It follows from [4, Corollary 2.3] that the structure (M, D) has no first-order reducts (up to interdefinability over ∅) other than itself and (M, =); that is, any structure on M which is ∅-definable in (M, D) is interdefinable over ∅ with (M, D) or (M, =).Since (M, E) is ∅-definable in (M, D) and is not a complete or null hypergraph, it is interderdefinable with (M, D), giving the claim.(Formally, [4,Corollary 2.3] describes the reducts of the structure (M, C) from Section 4.2 above, and shows that the only proper non-trivial reduct is (M, D), which is in [4] denoted (L, Q)).
To show that M 6 is set-homogeneous, using the homogeneity of (M, D) it suffices to prove the following claim.
Claim 2. If U, V ⊂ M are finite and σ : (U, E) → (V, E) is an isomorphism of the induced subhypergraphs, then the structures (U, D) and (V, D) are isomorphic.
Proof of Claim.Let A be a complete subhypergraph of U of maximal size, and put B = σ(A).It is easily seen that the D-structure on A = {a 1 , . . ., a n+2 } has the form depicted in Figure 11.Likewise B = {b 1 , . . ., b n+2 } carries a D-structure as depicted.We do not claim that σ(a i ) = b i for each i.
For each i = 3, . . ., n, let T i = {e : a 1 a 2 ea i a n+1 a n+2 is a non-edge}.Clearly T i is as depicted in Fig. 11.By maximality of A, we have U \ A = (T i : 3 ≤ i ≤ n), and maximality of |A| yields that For each e ∈ U \ A, write e ∼ a i if (A \ {a i }) ∪ {e} carries a complete hypergraph.It can be checked that if e ∈ T 3 then e ∼ a i for each i ∈ {1, 2, 3} and if e ∈ T n then e ∼ a i for i ∈ {n, n+1, n+2} but if e ∈ T i where 4 ≤ i ≤ n−1 then {j : e ∼ a j } = {i}.We say that e ∈ U \ A is peripheral if |{i : e ∼ a i }| = 3, and that e is central otherwise.We define T ′ i correspondingly in V (with each b i replacing a i ) and central and peripheral in the same way.Let P, P ′ be the sets of peripheral vertices of U, V respectively.Clearly |P | ≤ 2 and σ(P ) = P ′ , and σ induces some permutation π of {4, . . ., n − 1} such that if e is central in U then e ∈ T i if and only if σ(e) ∈ T ′ π(i) .Relabelling the b i if necessary (via a map i → n + 2 − i) we may suppose that σ(T Consider now the C-relation C U induced on U \R 3 with downwards direction towards a 1 ; for x, y, z ∈ U \ R 3 we have C U (x; y, z) ⇔ D(a 1 , x; y, z).Similarly let C V be the C-relation induced on V \ R ′ 3 with downwards direction towards b 1 .Observe that for any distinct u 1 , u 2 , u 3 , u 4 ∈ U \ R 3 , and any distinct x, y ∈ R 3 and distinct x ′ , y ′ ∈ R 3 , xyu 1 u 2 u 3 u 4 is a hypergraph edge if and only if x ′ y ′ u 1 u 2 u 3 u 4 is a hypergraph edge.Thus, a canonical 4-hypergraph structure is induced on U \R 3 and its complement is derived from the relation C U on this set as in Proposition 4.2.1.Likewise, a canonical 4-hypergraph is induced on V \R ′ 3 , and is derived from C V .Since σ induces an isomorphism of the 4-hypergraph on U \ R 3 onto that on V \ R ′ 3 , it follows by the proof of Proposition 4.2.1 that the corresponding structures (U \ R 3 , C U ) and (V \ R ′ 3 , C V ) are isomorphic, and hence that U and V carry isomorphic D-substructures, yielding Claim 2 and hence the result.Proof.Let K = Aut(N, E).Then K is a 5-homogeneous permutation group which is not 3-primitive.If K is not 3-transitive on N , then by Theorem 2.2.2(iii)K preserves a circular order or linear betweenness relation, and hence preserves a separation relation on N , as the latter is definable without parameters in a circular order or linear betweenness relation.So we may suppose that K is 3-transitive.It follows by Theorem 2.2.2(iv) that, assuming K does not preserve a separation relation on N , then there is a K-invariant D-relation D on N such that (N, D) is isomorphic to the 3-branching D-set (M, D) from the proof of Proposition 4.3.1.Since Aut(N, D) has two orbits on 6-sets, the result follows (one orbit gives a hypergraph isomorphic to M 6 , and the other gives the complement).✷ For completeness we also record the following immediate corollary of Theorem 2.2.2(iii), which may make applicable methods similar to those of Lemma 3.2.5.Part (v) of Theorem 2.2.2 also has consequences for set-homogeneous k-hypergraphs for k ≥ 9.
Corollary 4.3.3.Let k ≥ 5 and let N be an infinite set homogeneous khypergraph whose automorphism group is not 3-transitive.Then there is an ∅-definable (so Aut(N, E)-invariant) linear betweenness relation, or circular order on N .

Finite set-homogeneous structures, further questions
The literature on set-homogeneity is not well-developed.In this section explore briefly what can be said about finite set-homogeneous hypergraphs, and then pose some tentative conjectures and questions, concerning how far the notion extends beyond homogeneity.

The finite case
The finite homogeneous 3-hypergraphs were classified by Lachlan and Tripp in [32].There are just four examples, with automorphism groups PGL 3 (2), PGL 3 (3), PSL 2 (5), and the extension of PSL 2 (9) by an involutory field automorphism, each in the natural action on the projective plane or line.By homogeneity, the automorphism group of any homogeneous 3-hypergraph is 2transitive, and via the classification of finite simple groups all finite 2-transitive groups are known, making the Lachlan-Tripp result feasible.We have not attempted to carry out the corresponding classification under set-homogeneity, but first note that such hypergraphs again have 2-transitive automorphism group by the following result.Finite k-homogeneous groups which are not k-transitive (for some k ≥ 2) were classified by Kantor in [28], but this is not needed for the next result.Proof.This follows immediately from Lemma 3.2.3 for k = 3, and from Theorem 2.2.2(i) (and the rigidity of finite linear orders) for larger k. (As stated Theorem 2.2.2(i) has an assumption that X is infinite, but the proof in [11] only uses this to ensure that G is 2-homogeneous, which holds here anyway.) Enomoto [19] gave a very short proof that any finite set-homogeneous graph is homogenous.As noted in [25, Lemma 3.1], Enomoto's argument works also for finite tournaments, but not for finite digraphs (a directed 5-cycle is sethomogeneous but not homogeneous).The following example shows that his argument (at least in the original form) is not applicable to 3-hypergraphs.
The hypergraph M is not homogeneous, since it does not occur among the examples in [32].Indeed, the only 7-vertex example in [32] is the Fano plane, which has PSL 3 (2) as automorphism group.Since the latter is simple of order 168, it cannot have G as a subgroup -indeed, |G| = 42, and a group of order 168 with a subgroup of order 42 must have a proper normal subgroup of index at most 4!, so cannot be simple.
Proof of Theorem D. See Lemma 5.1.1 and Example 5.1.2.✷ Lachlan (see [30] and also [15] for the existence of a bound on rank) developed a very general structure theory for finite homogeneous relational structures.First, recall that a countably infinite structure M is smoothly approximable by a sequence M 0 ≤ M 1 ≤ . . . of finite substructures if M is ω-categorical, and (with G = Aut(M )), for any i ∈ N and tuples ū, v from M i , ū and v in the same Aut(M )-orbit if and only if they lie in the same orbit of the setwise stabiliser Aut(M ) {Mi} of M i .A rich theory around smooth approximation is developed in [14].
Roughly, the Lachlan theory says that if L is a finite relational language, then the finite homogeneous L-structures consist of finitely many 'sporadic' examples, and finitely many infinite families of examples, so that within each family the isomorphism type is determined by finitely many 'dimensions' taking values in N, the dimensions varying independently and freely above a certain minimum.The infinite 'limits' of these families are exactly the homogeneous countablyinfinite L-structures which are stable (see Section 5.2); they are 'smoothly approximated' by the finite families.It can be shown that something very similar holds under set-homogeneity, and we make this precise below.
Theorem 5.1.3.Let L be a finite relational language and let C be the collection of all finite set-homogeneous L-structures.Then we may write C = F 0 ∪ F 1 ∪ . . .∪ F t where F 0 is finite, and the structures in each F i (for i = 1, . . ., t) smoothly approximate a set-homogeneous countably infinite L-structure M i .
Proof.As L is fixed, set-homogeneity ensures that there is d ∈ N such that for each M ∈ C, Aut(M ) has at most d orbits on M 4 .The result now follows from Theorem 4.4.1 of [39], which is based on results from [14].It is almost immediate from the definition of smooth approximation that if a family F i smoothly approximates an infinite structure M i then M i will itself be set-homogeneous.

Infinite set-homogeneous structures
It seems feasible to classify set-homogeneous hypergraphs which are not thomogeneous for some small t.Problem 5.2.1.For k ≥ 3, classify set-homogeneous countably infinite (k + 1)-hypergraphs whose automorphism group is not k-transitive.In particular,
and both (b, c ′ 1 , . . ., c ′ r ) and (b, d ′ 1 , . . ., d ′ s ) are linear enumerations.Now since ¬R(c r ; c 1 , d 1 ) (as U is not linear), we have ¬R(c ′ r ; c ′ 1 , d ′ 1 ), so the clockwise angle from d ′ 1 to c ′ 1 at the centre is less than π, that is, d ′ 1 → c ′ 1 .Also, by considering angles at the centre, we have c ′ r + ∪ d ′ s − = T , and as c ′ 1 → c ′ r we have c ′ 1 ∈ c ′ r + .Likewise d ′ 1 ∈ d ′ s − (we allow 1 = s).It follows that we may choose b ∈ Z \ V with R(b; d ′ 1 , c ′ 1 ) and b ∈ d ′ s + ∩ c ′ r − , and for such b the extension of α with α(a) = b has the required properties.

Corollary 4 . 2 . 2 .
If (N, E) is a ≤5-set-homogeneous countably infinite 4-hypergraph whose automorphism group K is not 2-primitive, then either (N, E) is isomorphic to the structure M 4 of Proposition 4.2.1 or its complement, or K preserves a linear betweenness relation on N .Proof.The set-homogeneity assumption ensures that K is 3-homogeneous on N .Hence, by Theorem B(i), K is 2-transitive on N .Hence, by Theorem 2.2.2(ii),K preserves on N a linear betweenness relation or a relation C on N so that (N, C) is isomorphic to the structure (M, C) from Theorem 3.1.1.So suppose K ≤ Aut(N, C).Since Aut(N, C) has two orbits on 4-sets and (N, E) is ≤5-set-homogeneous, K also has the same two orbits on 4-sets of N .One such orbit gives a hypergraph isomorphic to M 4 , and the other gives its complement.We consider a further 4-hypergraph.Let (N, D) be the countably infinite dense proper homogeneous 4-branching D-set (see Section 2.1).Define a 4hypergraph N 4 = (N, E) from (N, D), whose edges are 4-sets which satisfy D under some ordering.

Corollary 4 . 3 . 2 .
Let (N, E) be a set-homogeneous 6-hypergraph whose automorphism group is not 3-primitive.Then either (N, E) or its complement is isomorphic to the structure M 6 from Proposition 4.3.1, or Aut(N, E) preserves a separation relation on N .
Proof of Theorem C. See Proposition 4.3.1 and Corollary 4.3.2.
and is strongly dense if (M, C) is proper as a C-set, (M, ≤) is a dense linear order without endpoints and in addition 2, 3, 4} with |J| = 4 − i, or P i * By Lemma 3.2.4,all of the formulas S 31 , S 41 , S 42 , S 13 , S 14 , S 24 must be realised, and it is routine by Lemma 3.2.5 and Table 1 to verify that (i) and (ii) are the only ways to realise all these S ij .Conditions P 2 {1,2} and P 2 {3,4} each realise three such formulas (namely S 31 , S 41 , S 42 , and S 13 , S 14 , S 24 respectively),

Table 1 :
0 in the (P k J , S ij )-entry means that if H satisfies P k J then it realises S ij .
These do not realize S 13 , S 31 , S 14 , S 15 , S 35 which need to be realized by P 3 and P 4 .The only possibilities of S 13 are: (a) P 4 T 1 T 2 T 3 T 4 S 13 S 31 S 14 S 41 S 15 S 51 S 24 S 42 S 25 S 52 S 35 S 53

Table 2 (
b) P 3 23 which needs S 31 , S 14 from P 4 P 3 35 , needs S 41 , S 14 from P 4 J , again impossible.4.M realises P 1 15J and P 2 15K .These do not realise S 13 , S 14 , S 15 , S 51 , S 52 , S 53 , which need to be realized by P 3 and P 4 .Again, the only possibilities for S 13 are: P 4 3 , P 3 23 , P 3 34 , P 3 35 .The first three need both S 15 and S 51 from the remaining condition, which is clearly impossible.P 3 35 needs S 14 , S 51 from P 4 J , again impossible. 5. M realises P 1 23J and P 2 23K .These do not realize S 31 , S 24 , S 25 , S 35 , T 2 which need to be realized by P 3 and P 4 .The only possibilities of S 31 are:(a) P41 , which needs S 24 , S 25 , T 2 from some P 3 These do not realize S 41 , S 24 , S 42 , S 25 , which need to be realized by P 3 and P 4 .The only possibilities of S 41 are: Proposition 4.2.1.Define a 4-hypergraph structure on M whose edge set E consists of 4-sets of form {x 1 , x 2 , y 1 , y 2 } such that C(x i ; y 1 , y 2 ) and C(y i ; x 1 , x 2 ) hold for i = 1, 2. Then M 4 = (M, E) is set-homogeneous and has 2-transitive but not 2-primitive automorphism group, so is not a homogeneous 4-hypergraph.