Around homogeneity

Around homogeneit y P eter J. Cameron Univ ersity of St Andrews T o the memory of Rob ert W o odrow Abstract F ort y-fiv e y ears ago, a young researc her in finite p erm utation group theory encoun tered a pap er b y Rob ert W o odrow. The homogeneous triangle-free graph W o o dro w describ ed there seemed to b e an infinite analogue of the Higman–Sims graph whic h had pla y ed an imp ortant role in the researc her’s thesis. The encounter changed the course of the researc her’s career. This paper is the story of that ev en t and its aftermath. The final section of the pap er suggests that F ra ¨ ıss ´ e classes of rigid structures are a p otentially interesting generalisation of Ram- sey classes. 1 In tro duction Rob ert W o o drow was a goo d friend. He grew up in rural Canada, and I in rural Australia, so w e had a common bac kground. W e had some mem- orable adven tures together, and on m y trip to Calgary to giv e the Louise and Ric hard Guy memorial lecture, he lo ok ed after me very well, taking me to Banff and also to the dinosaur-ric h Alb erta badlands. He is v ery m uch missed. But, although we only ha ve one joint paper [16], he had a very significan t effect on my mathematical dev elopment. His pap er [58] c haracterising Hen- son’s homogeneous triangle-free graph w as my in tro duction to homogeneous structures and F ra ¨ ıss ´ e theory . So I am grateful to him for launching me on this journey . In this paper I ha ve mostly surv eyed the directions I ha ve taken, with an emphasis on op en questions. 1 F or the remainder of this section, I provide some basic information ab out homogeneous relational structures. 1.1 Relational structures and homogeneit y A relational structure is a structure ov er a first-order language with no con- stan t or function sym b ols. In other w ords, w e hav e a n umber of “named” relations of v arious p ositiv e integer arities (the arit y of a relation is the num- b er of argumen ts it tak es). The commonest examples, each with a single binary relation, are graphs and (totally or partially) ordered sets. W e will meet others later. The underlying set is called the domain of the structure. An isomorphism b etw een relational structures ov er the same language is a bijection b etw een the domains whic h preserv es all the named relations; an automorphism of a structure is an isomorphism to itself. If M is a relational structure on a set X , and Y is a subset of X , the induc e d substructur e on Y has the set Y , and relations which are the restric- tions to Y of all the relations (in other words, we consider instances where all the arguments b elong to Y ). Note that w e forbid relations with no arguments. Such a relation would b e either true or false in any giv en structure, and structures giving it differen t truth v alues w ould not b e comparable. A structure M is homo gene ous if every isomorphism b et ween finite in- duced substructures of M can b e extended to an automorphism of M . Note that, in the early da ys, there was a confusion of terminology; some authors (including Rob ert W o o drow) used the term ultr ahomo gene ous for this con- cept, and reserv ed homo gene ous for the weak er concept that, if t wo finite induced substructures are isomorphic, then some isomorphism b etw een them can b e exteb ded to an automorphism of M . Note that the tw o notions here coincide for a sp ecial type of structure whic h I will consider later, those in whic h ev ery finite substructure is rigid (that is, has no non-trivial automor- phisms). An imp ortan t class of suc h structures consists of those for which one of the relations is a total order, since there is a unique total order on a giv en finite set and it is rigid. T o try to minimise confusion, I will use the term “weakly homogeneous” for the w eaker concept, and if necessary use “strongly homogeneous” for the stronger. I will discuss these further in the next section. A v ariant of homogeneity was devised by Jarosla v Ne ˇ set ˇ ril and me in 2004 [18]. A homomorphism f : A → B betw een relational structures is 2 a map which maps tuples satisfying a relation in A to tuples satisfying the same relation in B . (Th us, in graphs, edges must map to edges; but non- edges may map to edges, to non-edges, or to single vertices.) A structure A is said to ha ve the HH pr op erty , or to b e homomorphism-homo gene ous , if an y homomorphism b etw een finite induced substructures of A can b e extended to an endomorphism of A (a homomorphism from A to itself ). This concept has b een studied and extended by a num b er of authors [41, 22, 3]; but I will not discuss it further here, apart from a brief mention in Section 4. 1.2 Graphs and groups My notation for graphs and groups is relatively standard. The c omplete gr aph K n with n v ertices has all p ossible edges b etw een distinct v ertices, while the c omplete bip artite gr aph K m,n has t wo sets with m and n vertices, with all p ossible edges b etw een the tw o sets. A c omplete multip artite gr aph is similar but with any num b er of sets. The clique numb er of a graph is the largest num b er of v ertices in a com- plete induced subgraph, and the indep endenc e numb er is the largest n um b er of vertices in a set containing no edges. The line gr aph of a graph Γ is the graph L (Γ) whose v ertices are the edges (the pairs { x, y } satisfying the ad- jacency relation), tw o v ertices b eing joined if the corresp onding edges share a vertex. Groups will often b e automorphism groups of relational structures. I also need some terms from p ermutation group theory . P ermutations are written to the right of their argumen ts and comp osed left-to-right. Let G b e a p erm utation group on a set X . The orbit of an elemen t x ∈ X is the set of all images of x under elements of G . The group G is tr ansitive if there is a single orbit, that is, for any x, y ∈ X , there is an elemen t g ∈ G with xg = y . It is primitive if the only partitions preserv ed b y G are the “trivial” ones, the partition into singletons and the partition with a single part. F or a p ositiv e in teger k , the group is k -transitiv e if, giv en an y t wo k -tuples ( x 1 , . . . , x k ) and ( y 1 , . . . , y k ) of distinct elemen ts of X , there exists g ∈ G with x i g = y i for i = 1 , . . . , k . It is highly tr ansitive if it is k -transitive for all p ositive in tegers k . A p ermutation group is semir e gular if the stabiliser of an y point is the iden tity . (T op ologists call such an action fr e e .) A group is r e gular if and only if it is semiregular and transitiv e. A regular action of a group is isomorphic to the action on itself by right multiplication (the right r e gular action ). There 3 is also a left r e gular action , where g acts as left m ultiplication b y g − 1 ; it is isomorphic to the right regular action, and mak es a brief app earance in Subsection 7.2. Tw o elemen ts g , h of a group G are c onjugate if h = x − 1 g x for some x ∈ G . This is an equiv alence relation on G . Conjugate elemen ts ha ve the same order, and the same cycle structure if G is a p erm utation group. The Classific ation of Finite Simple Gr oups has had an enormous effect on the theory of finite p ermutation groups. F or just one instance, it sho ws that a 6-transitiv e finite p ermutation group m ust b e a symmetric or alternating group. (Things are different in the infinite case, as we shall see.) I will refer to it briefly as CFSG. 2 Finite homogeneous graphs The finite homogeneous graphs w ere classified by Gardiner [28]: Theorem 2.1 The finite homo gene ous gr aphs ar e as fol lows: (a) the disjoint union of c omplete gr aphs of the same size; (b) the c omplements of the ab ove (r e gular c omplete multip artite gr aphs); (c) the 5 -cycle; (d) the line gr aph of K 3 , 3 . Here is a brief sk etc h pro of. If Γ is homogeneous and v is an y v ertex, then the induced subgraphs of Γ on the neighbours and non-neigh b ours of v are b oth homogeneous. So we simply ha v e to show that there is no further homogeneous graph for which b oth of these subgraphs are on the list in the theorem, and are the same for any v . The theorem can b e refined as follows. W e say that a graph Γ is t - homogeneous (for a positive integer t ) if an y isomorphism b et w een induced subgraphs on at most t vertices can b e extended to an automorphism of Γ. There is also a combinatorial version of this. W e say that Γ is t -tuple r e gular if, given t -tuples ( x 1 , . . . , x t ) and ( y 1 , . . . , y t ) of vertices suc h that the map x i 7→ y i for i = 1 , . . . , t is an isomorphism of induced subgraphs, the num b er of common neighbours of x 1 , . . . , x t is equal to the n umber of common neigh b ours of y 1 , . . . , y t . (This notion is called t -isoregularity in 4 [46].) Th us 1-tuple regular graphs are regular, while 2-tuple regular graphs are strongly regular. The classification of 2-homogeneous finite graphs follo ws from the classi- fication of rank 3 p erm utation groups, a consequence of CFSG [34, 39, 40]. With the algebraic metho ds that had been used to study the 3-tuple regular graphs [14], I w as able to pro ve [7]: Theorem 2.2 A 5 -tuple r e gular finite gr aph is homo gene ous. There is more to the story . The Schl¨ afli gr aph on 27 v ertices is giv en b y Schl¨ afli’s “double-six construction” and describ es the 27 lines in a gen- eral cubic surface [50]. The McL aughlin gr aph on 275 v ertices was used by McLaughlin [42] to construct his sp oradic simple group. F or the Schl¨ afli graph, there are three 4-vertex subgraphs whic h do not o ccur as induced subgraphs: these are K 4 (one lab elling), K 3 ∪ K 1 (four lab ellings), and K 4 − e (six lab ellings). The remaining 53 lab elled graphs on four v ertices are single orbits of the automorphism group. So this graph is 4-homogeneous. The McLaughlin graph is 4-tuple regular, and contains ev ery 4-v ertex graph as an induced subgraph, but the indep endent sets of size 4 fall into t wo orbits under the automorphism group. So it is not 4-homogeneous (but this is the only failure of 4-homogeneity). F rom CFSG w e kno w that there are no further 4-homogeneous graphs. But the existence of 4-tuple but not 5-tuple regular graphs b eyond the Schl¨ afli and McLaughlin graphs is unkno wn. Question 2.1 Are there any further 4-tuple regular graphs? A further op en problem concerns recognition of these prop erties from the sp e ctrum of a graph, the eigen v alues and multiplicities of the adjacency matrix. A graph prop ert y P is determine d by the sp e ctrum if every graph whic h is cosp ectral with a graph having P also has P . Theorem 2.3 F or t  = 3 , the pr op erty of b eing t -tuple r e gular is determine d by the sp e ctrum. Question 2.2 Is it true that 3-tuple regularity is not determined b y the sp ectrum? 5 No w we can return to the question of w eak and strong homogeneit y , and note that they are the same for finite graphs. F or a graph which is weakly homogeneous is clearly t -tuple regular for all t , and so is homogeneous by Theorem 2.2. One can make a silly example where these concepts differ as follows. It is a simple exercise to show that ( Q , < ) (the rational n umbers with the usual order) is homogeneous. Now define a relation ρ on Q b y the rule that ρ ( x, y , z ) holds if x, y , z are all distinct and x < y . Then the order on any set of size at least 3 is determined by ρ ; but there are t wo ρ -isomorphisms b etw een any t wo sets of size 2, onlly one of whic h extends to an automorphism. Question 2.3 Is there a w eakly homogeneous infinite structure M for whic h, for every k , there is an isomorphism b et ween t wo k -elemen t substructures whic h do es not extend to an automorphism of M ? F or reasons of space I will sa y little ab out a more significan t generalisation of homogeneit y for finite graphs with which I ha ve recen tly b een inv olved [4]. A graph ∆ is an EPP A witness for a graph Γ if Γ is an induced subgraph of ∆ and ev ery isomorphism b et w een induced subgraphs of Γ can b e extended to an automorphism of ∆. (Th us a graph is homogeneous if and only if it is its o wn EPP A witness.) The name is an acron ym for “Extension Prop ert y for Partial Automorphisms”. Hrushovski [33] sho wed that every finite graph has a finite EPP A witness, but interesting questions concerning the smallest EPP A witness remain. The concept has imp ortant applications in sho wing significan t prop erties of infinite homogeneous graphs. 3 F ra ¨ ıss ´ e’s Theorem Clearly Gardiner’s metho d cannot work for finding countable homogeneous structures, since it is an induction on the cardinality of the structure. In fact this question had b een considered earlier b y Roland F ra ¨ ıss ´ e [27]. It was Rob ert W o o drow’s pap er which in tro duced me to F ra ¨ ıss ´ e’s metho d, and I so on realised what a p ow erful construction metho d it was. The metho d dep ends on recognising a structure M from its class of finite substructures (more precisely , the finite structures o ver the same language whic h are embeddable in M ). F ra ¨ ıss ´ e calls this class the age of M . Clearly the age of a coun table structure M is a class C of finite relational structures and has the following prop erties: 6 (a) It is closed under isomorphism. (b) It is her e ditary , that is, closed under taking induced substructures. (c) It con tains (at most) countably man y structures up to isomorphism. (d) It has the joint emb e dding pr op erty (JEP): any tw o structures in C can b oth b e embedded in a structure in C . In future I will use the term “class of relational structures” to include conditions (a), (b), (c) ab ov e. Note that (c) is automatic if the language has only finitely many relation sym b ols. It is not hard to see that the age of a coun table homogeneous structure has the amalgamation pr op erty (AP): given structures A, B 1 , B 2 ∈ C with em b eddings f i : A → B i for i = 1 , 2, there exists C ∈ C and embeddings g i : B i → C such that f 1 g 1 = f 2 g 2 , where f 1 g 1 means f 1 follo wed b y g 1 . Logicians normally forbid the empt y set as a structure. If w e allo wed the empty set, then (JEP) would b e a consequence of (AP), since we hav e forbidden relations of arity 0. F ra ¨ ıss ´ e’s theorem [27] asserts the following: Theorem 3.1 (a) A class C of finite structur es over a r elational language L (close d under isomorphism and induc e d substructur es and at most c ountable up to isomorphism) is the age of a c ountable homo gene ous r elational structur e M over L if and only if it also satisfies (JEP) and (AP). (b) If C satisfies these c onditions, then ther e is a unique c ountable homo- gene ous structur e M (up to isomorphism) whose age is C . A class C satisfying these conditions is a F r a ¨ ıss ´ e class , and the coun table homogeneous structure M is its F r a ¨ ıss´ e limit . F or example, it is straightforw ard to see that the class of all finite graphs is a F ra ¨ ıss ´ e class. Its F ra ¨ ıss ´ e limit is the sub ject of the next section. I end this section with a question. A p ermutation group G on a set X is oligomorphic if it has only finitely man y orbits on X n for all p ositiv e in tegers n . By the theorem of Engeler, Ryll-Nardzewski and Svenonius (Ho dges [32, Theorem 6.3.1], a coun table first-order structure is ℵ 0 -categorical (that is, sp ecified uniquely up to isomorphism among coun table structures b y its first- order theory) if and only if its automorphism group is oligomorphic; and in 7 this case, the automorphism group orbits on n -tuples are precisely the n - t yp es (where t w o n -tuples hav e the same type if and only if they satisfy the same n -v ariable first-order formulae). F urther detail can b e found in [32]. A homogeneous structure o ver a finite relational language is oligomorphic, since there are only finitely man y isomorphism t yp es of n -element substruc- tures for all n . The con verse, ho wev er, is false. Let V be a v ector space of coun table dimension ov er the 2-element field. Let X = V \ { 0 } and let G b e the group of all linear automorphisms of V . Then G is oligomorphic (for any finite set spans a finite subspace; G is transitiv e on subspaces of fixed finite dimension, and the stabiliser of suc h a subspace has only finitely man y orbits on its subsets). How ever, there is no homogeneous relational structure o ver a finite language whose automorphism group is G . This is b ecause, if k is the maximum arity of a relation in the language, and A and B are subsets of cardinality k + 1 such that A is linearly dep enden t and B satisfies the single linear relation asserting that the s um of its elements is zero, then A and B induce isomorphic relational structures but are not in the same orbit of G . Question 3.1 Is there a prop erty whic h distinguishes homogeneous struc- tures o ver finite languages among structures with oligomorphic automor- phism groups? A necessary condition is given b y the fact that the n umber of orbits on n - sets of a homogeneous structure ov er a finite relational language is b ounded ab o ve by the exp onential of a p olynomial; there is no suc h b ound for ℵ 0 - categorical structures. Ho wev er the b ound do es hold in the ab o v e example (the num b er of orbits on n -sets is roughly 2 n 2 / 4 ). 4 The Erd˝ os–R ´ en yi random graph, or Rado’s graph In 1963, Erd˝ os and R ´ en yi published a pap er on the lack of symmetry of finite random graphs. If a graph on n v ertices is formed by choosing edges indep enden tly at random (that is, b y tosses of a fair coin), then trivially ev ery n -v ertex graph occurs with non-zero probabilit y , and the probability of a giv en grap h is in v ersely proportional to the n umber of its automorphisms. 8 Sligh tly less trivially , the probabilit y that the random graph has an y non- iden tity automorphisms at all tends (rapidly) to 0 as n → ∞ ; and further, if the distance b etw een t w o graphs on the same v ertex set is measured b y the num b er of insertions and deletions of edges required to c hange one in to the other, then with high probabilit y a random graph lies at close to the maxim um p ossible distance from symmetry (that is, from a graph with a non-iden tity automorphism). By con trast, the pap er con tains a short tailpiece sho wing that a coun tably infinite random graph has infinitely many automorphisms almost surely . The real reason for this w as giv en a decade later in the b o ok [26] b y Erd˝ os and Sp encer on the probabilistic metho d. There is a particular coun table graph R which has the prop ert y that a random coun table graph is isomorphic to R almost surely . Moreo ver, R has infinitely man y automorphisms; indeed, it is homogeneous. The pro of is remark ably simple. Consider the “extension prop erty” that, giv en t w o disjoin t finite sets U and V of v ertices, there is a v ertex z joined to every v ertex in U and to none in V . Easy arguments sho w that the coun table random graph has this property almost surely . The bac k-and- forth metho d from mo del theory sho ws that tw o countable graphs with this prop ert y are isomorphic. Moreo ver, it is clear that the age of a graph with this prop ert y consists of all finite graphs. Also, giv en an isomorphism b et ween finite subgraphs, tak e tw o enumerations of the graph in whic h these finite subgraphs o ccur first, and then use back-and-forth to extend the isomorphism b et ween them to an automorphism. I will denote this graph by R , and call it the (countable) random graph. Erd˝ os and Sp encer sa y that this result “demolishes the theory of countable random graphs”; I would sa y that, in fact, it creates the new theory of the graph R . Erd˝ os and Sp encer gav e no explicit construction of R , since their argu- men t is a textb o ok example of a non-constructive existence proof (something whic h o ccurs with probability 1 certainly exists). But in the interv ening decade, Rado [48] had giv en an explicit construction. The v ertex set is the set N of natural num b ers. Given t w o natural n umbers x and y , we may as- sume that x < y ; then join x and y if the x th digit in the base-2 expression for y is 1. It is a simple exercise to verify the “extension prop erty” for this graph. Note that, if instead w e take this relation to b e directed, from smaller to larger, w e obtain a mo del of her e ditarily finite set the ory , that is, Zermelo– 9 F raenkel set theory with the Axiom of Infinit y replaced b y its negation. More generally , if w e take a coun table model of the ZF C axioms Empty Set, P airing, Union and F oundation, and ignore directions, the undirected graph w e obtain is R . Question 4.1 A t what p oint b etw een 1963 and 1974 did Erd˝ os realise the result whic h now app ears in [26]? Did Rado kno w ab out the result of Erd˝ os and R´ en yi, or vice versa? Another explicit construction of R is the follo wing. The vertex set is the set P 1 of primes congruen t to 1 (mo d 4); join p to q if q is a quadratic residue mo d p . By quadratic recipro city , the graph is undirected; using the Chinese Remainder Theorem and Diric hlet’s theorem on primes in arithmetic progressions, we can verify the extension prop erty which c haracterises R . The graph R w as a rich field for study . One of the first questions I ask ed m yself w as, do es it hav e an automorphism p erm uting all the v ertices in a single cycle? If there is suc h an automorphism σ , w e can lab el the v ertices with the in tegers so that σ acts as the cyclic shift. Let S b e the set of p ositiv e neigh b ours of the vertex 0. Then the whole graph is determined b y S : we ha ve x joined to y if and only if | y − x | ∈ S . I w as able to sho w that, if w e c ho ose S at random b y tossing a fair coin, then the graph we obtain is isomorphic to R almost surely . Moreo ver, t wo cyclic automorphisms of R are conjugate in the full automorphism group if and only if the corresponding sets S are equal. This demonstrates that Aut( R ) has 2 ℵ 0 conjugacy classes of cyclic automorphisms; so in particular, its cardinalit y is 2 ℵ 0 . There is another approac h to this. The coun table graph is sp ecified b y a sequence of 0s and 1s (for non-edges and edges); we can regard the set of all suc h sequences as a probability space. But we can also regard it as a metric space, where the distance b etw een t wo sequences is 1 / 2 n if they first disagree in the n th p osition. The metric space is complete, and so the notion of Baire category gives a second interpretation of “almost all”: the large sets are r esidual , that is, they contain a coun table in tersection of dense op en sets. (The Baire category theorem asserts that residual sets are non-empt y , and indeed they meet any op en set non-trivially .) The set of graphs isomorphic to R is residual in the space, again giving a nonconstructiv e existence pro of. On the face of it, probabilit y is more informativ e, since sets may hav e probability b etw een 0 and 1. But Baire category provides a to ol whic h is m uc h easier to use. In the space of all 10 binary sequences, a set S is op en if and only if it is finitely determine d (that is, an y mem b er of S has an initial finite sequence all of whose con tin uations lie in S ) and dense if and only if it is always r e achable (any finite sequence is an initial subsequence of a member of S ). A residual set contains an in tersection of countably many sets with these prop erties. No w it is easy to see that the “extension prop ert y” ab ov e for fixed U and V is finitely determined and alw a ys reac hable, and there are only coun tably man y c hoices for U and V ; so graphs isomorphic to R form a residual set of all graphs on a coun table vertex set. I conclude with t w o prop erties of R from [18]. A graph Γ is a sp anning sub gr aph of a graph ∆ if they hav e the same vertex set and every edge of Γ is an edge of ∆. Theorem 4.1 (a) A c ountable gr aph Γ c ontains R as a sp anning sub gr aph if and only if any finite set of vertic es of Γ has a c ommon neighb our. (b) A gr aph c ontaining R as a sp anning sub gr aph is homomorphism-homo gene ous. 5 Coun table homogeneous graphs Rob ert W o o drow’s pap er [58] shows that there are precisely four countable homogeneous graphs without triangles: a coun table null graph; a coun table disjoin t union of edges; a complete bipartite graph with t wo coun tably infinite parts; and one further example con taining all finite triangle-free graphs. This graph can b e explained as the F ra ¨ ıss ´ e limit of the class of all finite triangle- free graphs. In fact, this last graph was the first of an infinite family of countable homogeneous graphs. F or k ≥ 3, Henson ’s gr aph H k is the F ra ¨ ıss ´ e limit of the class of all graphs containing no induced subgraph K k (complete on k vertices). The pro of of the amalgamation property is simple: make the amalgam of B 1 and B 2 o ver A by making no iden tifications other than the v ertices in A and putting no extra edges b etw een B 1 and B 2 . (Henson’s construction [30] was different: he built his graphs inductively inside R .) A little later, W o o dro w with his do ctoral adviser Alistair Lachlan gav e a complete list [38]: Theorem 5.1 A c ountable homo gene ous gr aph is one of the fol lowing: 11 (a) A disjoint union of c omplete gr aphs of the same size, wher e either the numb er of p arts or the size of the p arts (or b oth) is c ountably infinite. (b) The c omplement of a gr aph in (a). (c) A Henson gr aph H k for k ≥ 3 . (d) The c omplement of a gr aph in (c). (e) The gr aph R . Sev eral further classifications of homogeneous relational structures ha ve b een found, but this stands as probably the most significant. As I ha v e said, W o o dro w’s c haracterisation of H 3 particularly appealed to me. In m y do ctoral thesis, the graph used b y Higman and Sims to construct a new sp oradic simple group [31] pla ys a starring role. At the time, we called it the Higman–Sims graph; but it was p ointed out later that it had b een constructed some time earlier by the statistican Dale Mesner [43], and later prov ed unique by him [44] (though he did not think to examine its automorphism group). This graph has the prop erties that its automorphism group acts primi- tiv ely on v ertices, and is transitive on ordered edges and ordered non-edges; the graph has no triangles, and the stabiliser of a v ertex is 3-transitiv e on the set of neigh b ours of that v ertex. F or comparison, the automorphism group of H 3 is primitiv e on vertices and transitive on ordered edges and ordered non-edges; the stabiliser of a vertex is highly tr ansitive (that is, k -transitiv e for ev ery p ositive in teger k ) on the set of neigh b ours of that vertex. This highly transitiv e group was lik e none I had seen b efore, and I w as inspired to lo ok for further examples. 6 B-groups Wielandt [56, Definition 25.1] defined a B-gr oup to b e a group with the fol- lo wing prop erty: an y primitive p ermutation group whic h contains the group G in its right regular action m ust b e 2-transitiv e. (In other w ords, if w e add p ermutations to kill all the non-trivial G -inv ariant equiv alence relations, then we kill all non-trivial G -inv arian t binary relations.) The groups w ere named for William Burnside, who pro ved that a finite cyclic group of order a prop er pow er of a prime n umber has this prop ert y . 12 (W e cannot call these groups Burnside gr oups , since this term w as already established for finitely generated infinite groups of finite exp onent.) The study of these groups w as v ery influen tial, and led to imp ortan t dev elopments in the theory of Sch ur rings. Ho w ever, the Classification of Finite Simple Groups completely c hanged the picture. F or example, w e kno w that for almost all natural num b ers n (all but O ( x/ log x ) num b ers b elo w x ), the only primitive groups of degree n are the symmetric and alternating groups, and so every group of order n is a B-group [19]. Ho wev er, the picture in the infinite case is very differen t. Ken Johnson visited Oxford in 1981–2 and asked whether there are any countably infinite B-groups. Graham Higman w as able to give a rather general condition en- suring that a countable group was not a B-group. Then Johnson and I [15] sho wed that ev ery group satisfying a w eaker form of Higman’s condition is a regular subgroup of the automorphism group of R . Since Aut( R ) is primitiv e but not 2-transitive, all groups satisying this condition fail to b e B-groups. Here is a statement of our theorem. In a group G , a squar e-r o ot set is a set of the form √ a = { x : x 2 = a } for a fixed element a ; it is non-princip al if a  = 1. Theorem 6.1 L et G b e a c ountable gr oup which c annot b e written as the union of finitely many tr anslates of non-princip al squar e-r o ot sets to gether with a finite set. Then G is emb e dde d as a r e gular sub gr oup of Aut( R ) . F or an y finite or countable group G , the group G × C ∞ satisfies the h yp otheses of this theorem. Thus Aut( R ) embeds ev ery finite or coun table group as a semiregular subgroup. As far as I know, no countable group has ever b een sho wn to be a B-group. The b est candidate I kno w is the infinite dicyclic group ⟨ x, y | y 4 = 1 , y − 1 xy = x − 1 ⟩ whic h fails to satisfy our v ersion of Higman’s condition: it can b e written as p y 2 ∪ y p y 2 . Question 6.1 Is there a countable B-group? 13 7 Homogeneous Ca yley ob jects Although I had to lea ve the question of countable B-groups unresolv ed, fur- ther w ork on it let me to in teresting mathematics and collab orations with t wo remark able mathematicians. The use of R in the preceding section suggested turning the question on its head. Let M b e a countable structure. I will say that M is a Cayley obje ct for the countable group G if its p oin t set can b e identified with G so that the righ t regular action of G is contained in the automorphism group of M . This is motiv ated by the notion of Cayley gr aph , a graph whic h is a Ca yley ob ject for a group G . In tro duced b y Cayley in the 19th cen tury , these graphs are no w fundamental ob jects in geometric group theory . My hop e w as that, b y c ho osing homogeneous structures whose automorphism group w as primitiv e but not 2-transitiv e and expressing them as Ca yley ob jects for suitable groups, I w ould find further non-B-groups. Although I had no success with this, the search did lead me into some v ery in teresting byw a ys, some of which I will now describ e. 7.1 Hyp ergraphs F or k > 2, there is a unique countable homogeneous universal k -uniform h yp ergraph H . It can b e shown that it is a Ca yley ob ject for ev ery countable group. How ever, this giv es no further B-groups, since the automorphism group of H is ( k − 1)-transitive. (Thus all degrees of transitivit y are realised b y infinite p ermutation groups.) 7.2 Sum-free sets Of course, one of the first examples I looked at w as H 3 , Henson’s triangle-free graph. I lo ok ed first at cyclic automorphisms. Henson [30] had sho wn in his original pap er that the graph do es admit cyclic automorphisms. F ollowing the approac h that w orked for R , what w as needed w as to c haracterise the sets S of p ositive integers such that the graph Γ( S ) obtained by joining tw o in tegers x and y if | y − x | ∈ S is isomorphic to Henson’s graph. It is easy to see that Γ( S ) is triangle-free if and only if S is sum-fr e e , that is, do es not con tain x, y , z with x + y = z . (W e allo w x = y here.) So the question w as: do almost all sum-free sets S pro duce Henson’s graph? It is not hard to write 14 do wn a condition on S for Γ( S ) to b e Henson’s graph; suc h sum-free sets are called universal . I tried and failed to find a nice isomorphism-in v arian t measure to do the job. Later, by completely differen t techniques resem bling graphon theory , P etrov and V ershik [47] found suc h a measure, and this w as extended to a m uch wider class of structures b y Ac k erman, F reer and Patel [1]. Ho wev er, Baire category was able to do the job. It is not hard to sho w that the set of univ ersal sum-free sets is residual in the set of all sum-free sets. The explicit construction of suc h a set inv olv ed leaving increasingly long gaps, and I conjectured that a universal sum-free set has densit y zero. This conjecture was confirmed b y Schoen [51]. There is a nice pattern her e. Two old results related to Ramsey’s theorem are: • V an der W aerden’s theorem [55]: in a partition of N in to finitely man y parts, one part contains arbitrarily long arithmetic progressions. • Sc hur’s theorem [52]: in a partition of N into finitely many parts, one part contains tw o num b ers and their sum. F amously , v an der W aerden’s theorem w as strengthened by Szemer ´ edi [53] to the statement that a set of natural n um b ers of p ositive upp er densit y con tains arbitrarily long arithmetic progressions. But the analogous strengthening of Sc hur’s theorem is false: the set of o dd num b ers is sum-free but has density 1 2 . The results noted ab o ve give a kind of replacement: almost all sum-free sets in the sense of Baire category (the universal ones) are excluded by the p ositiv e upp er density condition. Sum-free sets pro v ed to b e a fascinating topic. Of course, I wan ted to coun t them, and this problem whic h I p osed in 1987 [8] caugh t the interest of P aul Erd˝ os, and led to m y first join t pap er with him [13]. The conjecture w as that the n um b er of sum-free subsets of { 1 , . . . , n } is asymptotically c e 2 n/ 2 if n is ev en and c o 2 n/ 2 if n is odd, where c e and c o are t w o constan ts (whose v alues are roughly 6 . 8 and 6 . 0). More precisely , the conjecture asserts that almost all suc h sets either consist of o dd n umbers or hav e all entries in ( n/ 2 − w ( n ) , n ], where w ( n ) is a function gro wing arbitrarily slo wly . W e were able to sho w that for these tw o types our asymptotic was correct. Later Green [29] and Sap ozhenk o [49] prov ed the conjecture, b y sho wing that the num b er of sets not of these tw o types w as asymptotically smaller than 2 n/ 2 . 15 Question 7.1 Estimate the constan ts c e and c o more precisely . Are they transcenden tal? There is a probabilit y measure defined on sum-free sets of natural num b ers as follows. Start with the empt y set, and consider the natural n umbers in turn. If n = x + y where x, y ∈ S , then n / ∈ S ; otherwise include n in S b y the toss of a fair coin. This measure has some in teresting prop erties but little is known ab out it. Here is a sample question, suggested by exp erimen t. Question 7.2 Examine this space further. Is it true that a random sum-free set constructed as ab o ve has a densit y almost surely , and that the sp ectrum of densities is discrete ab o v e 1 6 but has a contin uous part b elo w this v alue? It is kno wn, for example, that the probability that a random sum-free set con tains no even num b ers is non-zero (it is approximately 0 . 218), and that conditioned on this, the density is almost surely 1 4 . Other types of sum-free set (for example, those contained in the residue classes 1 and 4 (mod 5), or those in whic h 2 is the only ev en n umber) also ha ve p ositive probabilit y [9, 5]. Henson’s graph H 3 is a Cayley graph for v arious countable groups, but none which are not already accounted for b y the random graph R . What ab out the higher Henson graphs? Henson had sho wn that, for k ≥ 4, the graph H k has no cyclic automorphism. I extended this to show that it is not a normal Cayley gr aph for an y countable group. (A normal Ca yley graph is a graph Ca y( G, S ) for which S is a normal subset of G , that is, fixed by conjugation; equiv alen tly , the graph admits b oth the righ t and the left regular actions of G .) In particular, H k is not a Ca yley graph for any ab elian group if k ≥ 4. Then Cherlin [21] sho w ed that these graphs are Cayley graphs for the free group of countable rank, and also for the free nilp otent group of class 2 and coun table rank. His tec hniques also w ork for a wider class of graphs, including some related to the Urysohn space, as well as for v arious directed graphs and metric spaces. 7.3 The Urysohn space After I talked ab out R at the Europ ean Congress of Mathematics in Barcelona in 2000, Anatoly V ershik in tro duced himself to me and told me ab out the 16 Urysohn sp ac e , the universal homogeneous Polish space (complete separable metric space). It is to o big to b e the F ra ¨ ıss ´ e limit of finite metric spaces, but as Urysohn realised in the 1920s, there is a simple w ay around this. A r ational metric sp ac e is a metric space in which all the distances are rational n umbers. There are only coun tably many rational metric spaces, and they do form a F ra ¨ ıss ´ e class; their F ra ¨ ıss ´ e limit is the r ational Urysohn sp ac e , and the usual pro cess of completion now gives the Urysohn space. With V ershik’s guidance, and the exp erience of dealing with other F ra ¨ ıss´ e classes, we were able to prov e some striking results ab out isometries of Urysohn space, esp ecially ab elian groups acting regularly on it. (This is done by finding a cyclic automorphism of the rational Urysohn space; its or- bits on the completion are dense, so its closure is a transitiv e ab elian group. These give a num b er of w ays of putting an ab elian group structure on the Urysohn space (although the constructions are not explicit). See [20]. Question 7.3 Noting that the completion pro cess may in tro duce torsion, w e can ask: can we sa y an ything ab out the torsion subgroups of the ab elian groups which arise? In particular, is there a torsion-free transitive ab elian subgroup? 7.4 Multiorders A little though t sho ws that the structure ( Q , < ) (the rational n umbers as ordered set) is homogeneous (its age is the set of all finite total orders), and is a Cayley ob ject for the additiv e group of Q . Indeed, this is the example whic h F ra ¨ ıss´ e generalised in his pap er [27]. A multior der is simply a set carrying a num b er of total orders. If there are m orders, we sp eak of an m -order. A finite 2-order on n points is simply a p ermutation of { 1 , . . . , n } , re- garded as a reordering of these lab els, not as a function on the set of lab els. The first order can b e used to label the n p oints with 1 , 2 , . . . , n ; then the second order rearranges the lab els. No w a substructure of a 2-order (as rela- tional structure) gives a p erm utation on { 1 , . . . , k } in whic h the lab els come in the same order as in the p ermutation on { 1 , . . . , n } . F or example, [1 , 3 , 2] is a substructure of [2 , 4 , 1 , 3], using the first, second and fourth entries. This is exactly the notion of subp ermutation whic h o ccurs in the theory of p er- m utation patterns [57]. 17 The finite m -orders form a F ra ¨ ıss´ e class. I in vestigated its F ra ¨ ıss´ e limit as a Ca yley ob ject [12]. All I w as able to pro ve was that the homogeneous m - order is a Ca yley ob ject for the free ab elian g roup ( C ∞ ) n if and only if m < n . The pro of required Kronec ker’s theorem on Diophantine appro ximation [36]. F or example, take the elemen ts of ( C ∞ ) 2 as p oin ts of the plane with integer co ordinates. Now to construct a dense total order on this set, tak e a line with irrational slop e, and slide it across the plane. The order of the p oin ts is given b y the order in which they are hit by the sliding line. T ranslations of the set b y v ectors with integer co ordinates preserv e the order. A group is a Cayley ob ject for a total order if and only if it is right or der e d ; for the order to b e homogeneous we require the right order on the group to b e dense. Question 7.4 Find other examples of groups for which the univ ersal m - order is a Cayley ob ject, for m > 1. 7.5 Miscellanea There are interesting graphs which are in a sense “almost homogeneous”: w e can pro duce a homogeneous structure by adding a relation. One example is the generic bip artite gr aph : this is not homogeneous, since non-adjacen t v ertices ma y b e at distance 2 or 3, but if we add an equiv alence relation to the language and interpret it as the bipartition, we obtain a homogeneous structure. But its automorphism group is imprimitiv e, so we obtain no B- groups. A more in teresting example is Cov ington’s N-free graph [24]. A graph is N-fr e e , or a c o gr aph , if it con tains no path of length 4 as induced subgraph. If w e hav e a 3-clique C in suc h a graph, it is easy to see that if t w o vertices v , w are eac h joined to a single vertex in C , then it must b e the same v ertex. Thus, w e need a relation to pick out one of the three vertices as a p otential single neigh b our of a p oint outside. A similar remark holds for indep endent sets of size 3. Any other 3-vertex subgraph already has a distinguished vertex. Co vington sho wed that by adding a ternary relation whic h distinguishes one of its three argumen ts suitably , we obtain a homogeneous structure. This structure is a Cayley ob ject for the coun table elemen tary ab elian 2-group. This ternary relation is called a C-r elation ; it plays a role in the classi- fication of Jordan groups of countable degree [2], and will crop up again in the last section of this pap er, where I will give a description of it. 18 Question 7.5 Is it true that a group for which Co vington’s graph is a Cayley ob ject m ust b e a 2-group? Are there examples other than the elementary ab elian group? 8 Reducts and o v ergroups Let M b e a homogeneous relational structure. A structure N on the same p oin t set is a r e duct of M if the relations of N ha ve first-order definitions without parameters in M . Reducts are defined up to equiv alence, tw o struc- tures equiv alen t if eac h is a reduct of the other. As an example, here are the reducts of the structure ( Q , < ). This list is a consequence of my first pap er on infinite structures [6], though I did not know at the time that I was pro ving that. (a) The structure ( Q , < ). (b) The b etwe enness r elation on Q , the ternary relation β , where β ( x, y , z ) if and only if x < y < z or z < y < x . (c) The cir cular or der γ , the ternary relation whic h holds if and only if x < y < z or y < z < x or z < x < y . (d) The sep ar ation r elation σ , the quaternary relation for which σ ( w , x, y , z ) holds if and only if w and y are b et w een x and z and vic e versa . (e) The pure set Q with no relations. There is a natural topology on the symmetric group, the top ology of p oin twise con v ergence; a basis for the op en neigh b ourho o ds of the identit y is giv en b y the p oin twise stabilisers of finite tuples. In this topology , a subgroup of the symmetric group is closed if and only if it is the automorphism group of a relational structure, which can b e taken to b e homogeneous (but w e cannot assume that the language is finite). Now it is clear that, if N is a reduct of M , then Aut( N ) is a closed o vergroup of Aut( M ), and conv ersely (for homogeneous structures ov er a finite language). I suggested a list of reducts of R , and Simon Thomas [54] pro v ed that it w as complete. I will describ e the groups rather than the structures. (a) Aut( R ). 19 (b) The group of automorphisms and anti-automorphisms (isomorphisms to the complement) of R . (c) The group of switc hing automorphisms of R . (Switc hing is the op era- tion of a graph given by c ho osing a subset Y of the v ertex set X , and exc hanging edges and non-edges b etw een Y and its complemen t, keep- ing the induced subgraphs on Y and on X \ Y as they w ere; a switc hing automorphism is a bijection on the vertex set which maps the graph to an image under switching.) (d) The group of switching automorphisms and anti-automorphisms of R . (e) The symmetric group on the v ertex set of R . Note the structural similarity with the reducts of ( Q , < ). In particular, the degree of transitivit y of the reducts (a)–(d) is 1, 2, 2, 3, and (a) and (c) are normal subgroups of (b) and (d) of index 2, in each case. This result led Thomas to conjecture that a homogeneous structure o ver a finite relational language has only finitely man y reducts. This has b een verified in a num b er of cases. Question 8.1 Decide Thomas’ conjecture. The automorphism group of a homogeneous relational structure may ha v e man y o vergroups in the symmetric group whic h are not closed, and hence not reducts. F or example, the finitary symmetric group F on a coun table set X (the group of permutations which mo v e only finitely man y points) is a normal subgroup of the symmetric group, so for any subgroup G , the pro duct F G is a subgroup; it is highly transitive, so its closure is the symmetric group. But some of these ov ergroups are in teresting in their o wn righ t. My only join t pap er with Rob ert W o o drow came ab out when he and his colleagues Claude Laflamme and Maurice P ouzet w ere examining automorphism groups of h yp ergraphs (with infinite hyperedges, so not relational structures: for example, the h yp ergraph H whose edges are the subsets inducing copies of R ), while Sam T arzi and I were lo oking at homeomorphism groups of filters or top ologies (for example, the neigh b ourho o d filter of R , generated by the v ertex neighbourho o ds in R ), and groups of p ermutations which are “almost automorphisms” of R . W e com bined forces to pro duce the pap er [16]. The pap er contains a couple of op en questions ab out the relationships b etw een these groups. 20 Examples of groups of “almost automorphisms” include Aut 1 ( R ), the group of p ermutations changing only finitely man y adjacencies (edges to non-edges or vic e versa ); Aut 2 ( R ), the group of p erm utations changing only finitely many adjacencies at each vertex; and Aut 3 ( R ), the group of p ermu- tations changing only finitely man y adjacencies at all but p ossibly finitely man y v ertices. The pap er con tains a couple of op en questions ab out the relationships b et ween these groups. F or example, let F Aut( H ) denote the group of p erm u- tations g suc h that there is a finite subset S of R suc h that, for ev ery edge E of H , b oth ( E \ S ) g and ( E \ S ) g − 1 are edges of H . It is known that Aut( H ) · FSym( R ) ≤ F Aut( H ) , where FSym( R ) is the group of p erm utations of finite supp ort on the v ertex set of R , but it is not kno wn whether equality holds. Question 8.2 Resolv e the outstanding questions ab out equalities and in- clusions in [16]. The pap er also shows: Theorem 8.1 A n over gr oup of Aut( R ) which is not a r e duct must b e either c ontaine d in the gr oup B of switching automorphisms and anti-automorphisms, or highly tr ansitive. F or, if it is not contained in B , then its closure m ust b e the symmetric group. Question 8.3 Determine the ov ergroups of Aut( R ) con tained in B . An example is the group of switc hing automorphisms where the switc hing set is finite. 9 Ramsey classes A simple form of Ramsey’s theorem states that, given p ositiv e integers a and b , there exists c suc h that, if the a -element subsets of a c -elemen t set are coloured red and blue, there will b e a mono chromatic b -element set. 21 In the 1980s, interest grew in a structural v ersion of the theorem, where w e replace sets b y structures in a class C ov er a finite relational language. As usual, we assume our classes are closed under isomorphism and heredi- tary . W e also ha ve to replace substructures by em b eddings, since differen t em b eddings with the same image destro y the Ramsey prop ert y . Th us, C is a R amsey class if, giv en structures A, B ∈ C , there exists C ∈ C suc h that, if the em b eddings A → C are coloured red and blue, there exists an em b edding B → C such that all the embeddings A → C with image in B ha ve the same colour. Jarosla v Ne ˇ set ˇ ril [45] show ed tw o general prop erties of Ramsey classes. Theorem 9.1 L et C b e a R amsey class over a finite r elational language. Then (a) C is a F r a ¨ ıss ´ e class; (b) if the structur es in C ar e non-trivial (not pur e sets), then they ar e rigid (that is, they have trivial automorphism gr oups). Ne ˇ set ˇ ril found man y examples of Ramsey classes, and prop osed a pro- gramme to classify them: determine the F ra ¨ ıss ´ e classes, and then decide whic h of them hav e the Ramsey prop ert y . All of Ne ˇ set ˇ ril’s examples were rigid b ecause a total order was part of the structure: more formally , they hav e a total order as a reduct (whic h ma y b e done by imp osing a total order on all the structures in the family). A t the time, I constructed a F ra ¨ ıss ´ e class of rigid structures by a differen t metho d, and wondered whether or not it was a Ramsey class. The answer came in the 1990s with the theorem of Kec hris, P estov, and T o dorˇ cevi´ c [35]. It uses the top ology of p oin twise conv ergence on the sym- metric group, in tro duced in the preceding section. A top ological group G is said to b e extr emely amenable if, for an y con tin uous action of G on a compact space X , there is a p oin t of X fixed by G . Theorem 9.2 L et C b e a F r a ¨ ıss´ e class, with F r a ¨ ıss´ e limit M . Then C is a R amsey class if and only if Aut( M ) is extr emely amenable. This answers my question. The set of total orders on a coun table set can b e shown to b e compact, in a natural top ology , with the symmetric group acting con tin uously on it. So, if C is a F ra ¨ ıss ´ e class with F ra ¨ ıss´ e limit M , then 22 Aut( M ) fixes a total order. It follo ws from the Engeler–Ryll-Nardzewski– Sv enonius theorem of mo del theory that this total order is a reduct of M , and so can b e defined without parameters. Hence ev ery finite structure in M carries a total order, and thus is rigid. I will describ e m y example in the next section, where I will show that it do es not ha ve a total order as a reduct, and hence is not a Ramsey class. With Sia v ash Lashk arighouchani [17], I was able to find an explicit failure of the Ramsey prop erty in my class, and indeed in an y F ra ¨ ıss ´ e class of rigid structures whic h is not a Ramsey class. This failure in volv es structures A and B with | A | = 2. 10 F ra ¨ ıss ´ e classes of rigid structures This final section considers F ra ¨ ıss ´ e classes of rigid structures, whic h (as noted in the Introduction) are precisely those for whic h the w eak and strong v er- sions of homogeneit y (formerly called homogeneity and ultrahomogeneity) coincide. This is a class which migh t repay inv estigation, and I suggest some p oin ters. A F ra ¨ ıss ´ e class C is said to ha ve the str ong amalgamation pr op erty if, for an y amalgamation A → B 1 , B 2 → C , the structure C and the embeddings of B 1 and B 2 in to C can b e c hosen so that the intersection of the images of B 1 and B 2 is precisely the image of A (and not larger). This is equiv alen t to sa ying that, if M is the F ra ¨ ıss ´ e limit and G = Aut( M ), the stabiliser of a finite set F in G has no finite orbits outside F . In mo del-theoretic terms, this says that algebraic closure in M is trivial. Classes with the strong amalgamation prop erty are imp ortant in several areas. F or example, the theorem of Ac k erman et al. [1] on in v arian t measures sho ws that this condition is necessary and sufficient for their construction of an inv arian t measure. Also, if G is a p ermutation group on a countable set in whic h the stabiliser of a finite set has no finite orbits outside the set, then either G is highly transitiv e, or G preserv es a non-trivial top ology (and if primitiv e, it preserves a non-trivial filter) [11, Section 4]. If tw o F ra ¨ ıss ´ e classes C 1 and C 2 ha ve the strong amalgamation prop ert y , w e may form the class C = C 1 ⊙ C 2 o ver the union of the languages for the t wo classes: a structure in this class consists of a finite set with indep endently- c hosen structures from C 1 and C 2 . It is straightforw ard to see that C is also a F ra ¨ ıss ´ e class with strong amalgamation. 23 F or example, the class C m of m -orders can b e defined inductively by the rules that C 1 is the class of total orders and C m = C m − 1 ⊙ C 1 for m > 1. A tournament is a set with a binary relation τ suc h that τ ( x, x ) never holds, while for x  = y , exactly one of τ ( x, y ) and τ ( y , x ) holds. It is easy to see that the class of finite tournamen ts is a F ra ¨ ıss ´ e class with strong amalgamation. Its F ra ¨ ıss ´ e limit is the generic tournament , with prop erties resem bling those of the random graph; in particular, it can b e constructed using the set P − 1 of primes congruent to − 1 (mo d 4), with an arc from p to q if q is a quadratic residue mo d p . Moreo ver, if T is a finite tournament, then | Aut( T ) | is o dd: for, if it were ev en, then it would contain an element of order 2, whic h would interc hange t wo v ertices; but this is clearly not p ossible. The other ingredient is a C-relation, one of those inv olv ed in the classi- fication of infinite Jordan groups [2]. Let X b e the set of leav es of a ro oted binary tree. Define the ternary relation γ on X by the rule that γ ( x, y ; z ) holds if [ x, r ] ∩ [ y , r ] ⊃ [ x, r ] ∩ [ z , r ] = [ y , r ] ∩ [ z , r ] , where r is the ro ot, and [ a, b ] denotes the unique path from a to b . See Figure 1. r r r r r A A A A A A       A A A r x y z Figure 1: C-relation γ ( x, y ; z ) It can b e sho wn that the C-relation on the leav es determines the binary tree, and hence that the class of finite C-relations is a F ra ¨ ıss´ e class with the strong amalgamation prop erty . Moreov er, the automorphism group of any C-relation is a 2-group. F or such an automorphism m ust preserve the binary tree. An automorphism of o dd prime order fixes the ro ot, and hence its tw o c hildren, and so on down the tree; thus no such automorphism can exist. So the automorphism group of an y finite C-structure is a 2-group. 24 No w, if C 1 is the class of finite tournaments, and C 2 the class of finite C-relations, then C = C 1 ⊙ C 2 is a F ra ¨ ıss ´ e class of finite relational structures. These structures are rigid: for the only finite group whose order is b oth o dd and a p o wer of 2 is the trivial group. Since the structure of an y 2-set is an arc of the tournamen t, we see that the automorphism group of the F ra ¨ ıss ´ e limit is transitiv e on 2-element sets. Th us, there cannot b e a reduct which is a total order, since it would agree with the tournamen t on some but not all pairs, and the group would not b e transitiv e on 2-sets. Th us, by the theorem of Kechris et a l. , C is not a Ramsey class. Here is a direct pro of of that fact. First I state the negation of the Ramsey prop ert y: there exist A and B suc h that, for any C , there is a colouring of the em b eddings A → C red and blue so that there is no mono c hromatic copy of B . T o demonstrate this, w e tak e A to b e a 2-elemen t structure, and B a 3-elemen t structure for which the induced tournament is a 3-cycle. Giv en an y C , w e tak e a total ordering of the points of C , and colour the em b eddings A → C red if the total order agrees with the tournamen t arc on the image, blue if it disagrees. Then an y 3-cycle of the tournamen t on C will hav e b oth red and blue embeddings of A . As noted earlier, it is shown in [17] that there is an explicit failure of the Ramsey prop ert y in any F ra ¨ ıss ´ e class of rigid structures which do es not hav e a total order as a reduct; w e can tak e | A | = 2. The following question is raised: Question 10.1 If C is a F ra ¨ ıss ´ e class of rigid structures which does not ha v e a total order as a reduct, is there a failure of the Ramsey prop erty with | A | = 2 and | B | b ounded b y a function of the num b er of isomorphism types of 2-element structures? F or the rest of this section, I will b egin a study of F ra ¨ ıss ´ e classes of rigid structures, whic h migh t profitably be tak en further. Let C b e suc h a class and M its F ra ¨ ıss ´ e limit. Then Aut( M ) is a torsion-free group, since an element of finite order w ould p erm ute some finite set non-trivially . F rom no w on I also assume that M has no reduct which is a total order. Question 10.2 Are there examples where Aut( M ) is a simple group? No w here are a couple of observ ations. 25 Note 1 M has a reduct which is a tournament. F or, b y homogeneity , for an y 2-set { x, y } , there is a relation which orders the pair (else there w ould b e an automorphism exc hanging them); and the collection of all suc h relations defines a tournamen t structure. (T ak e the relations in turn, and put x → y if the first relation to order them satisfies this.) This tournamen t is vertex-transitiv e. W e might b egin the study by assum- ing that the tournamen t is homogeneous, since all homogeneous tournamen ts are known [37]. I will call a ternary relation whic h distinguishes one of its three argumen ts an AM-r elation , after Samuel T aylor Coleridge’s lines [23]: It is an ancient Mariner, And he stopp eth one of three. A C-relation is an example. Note 2 M has a reduct which is an AM-relation. 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