Infinite Permutation Groups in Enumeration and Model Theory

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1 Infinite Permutation Groups in Enumeration and Model Theory Peter J. Cameron School of Mathematical Sciences, QMW, University of London, Mile End Road London El 4NS, UK 1. Introduction A permutation group G on a set Q has a natural action on Q n for each natural number n. The group is called oligomorphic if it has onlyfinitelymany orbits on Q 11 for all «GN. (The term means "few shapes". Typically our permutation groups are groups of automorphisms of structures of some kind; oligomorphy implies that the structure has onlyfinitelymany non-isomorphic w-element substructures for each n.) Oligomorphic permutation groups have close connections with both model theory and combinatorial enumeration. For the former, a basic result is the theorem of Engeler, Ryll-Nardzewski and Svenonius characterizing Ko-categorical countable structures as those whose automorphism groups are oligomorphic. The connection with enumeration is via homogeneous structures, those for which orbits on 77-sets are isomorphism types of induced substructures. A theorem of Fraïssé gives us a rich supply of homogeneous structures. These matters are described in Section 2. Section 3 develops some tools of enumeration theory (cycle index) in this context, with a few applications. In Section 4, the famous countable "random graph" of Erdös and Rényi is used to introduce the ideas of measure and Baire category. In the fifth section, some results and problems on the rate of growth of orbit-counting sequences are presented. Finally, the search for cyclic automorphisms of certain interesting graphs leads to sum-free sets, which have a fascinating theory. For an oligomorphic permutation group G, I let f n (G), F n (G) and F*(G) be the numbers of orbits of G onrc-sets,n-tuples of distinct elements, and all n- tuples respectively. By convention, /o(g) = Fo(G) = FQ(G) = 1. There are some interesting relations among these sequences, notably n Ft(G) = Y,S(n,k)F k (G), where S(n,fc)is the Stirling number of the second kind; this fact has a number of combinatorial consequences (Cameron and Taylor 1985). I will always assume that the set Q on which a group acts isfiniteor countable. (Little is lost here; the downward Löwenheim-Skolem theorem of model theory guarantees that any sequences (f n (G)), (F n (G)), etc. realized by an oligomorphic group can be realized by a group of countable degree.) Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990 rz\ -ri-,«iv/tchv.omo+; oi Qnr.;au, r^r To«a«1001

2 1432 Peter J. Cameron Notation: G x H and G Wr H denote the direct and wreath products of the permutation groups G and H (acting on the disjoint union and cartesian product respectively of the sets admitting G and H); G a is the stabilizer of the point a eü. S œ is the symmetric group of countable degree; C n, the cyclic group of order n; and A, the group of order-preserving permutations of Q. Note that f n (A) = 1 for all n: any order-preserving bijection between finite subsets of Q can be extended to a (piecewise-linear) order-preserving permutation of Q. Three properties of Q will be important. First of course is Cantor's (1895) characterization of Q as countable dense ordered set without endpoints. Second, as just mentioned, any order-preserving bijection between finite subsets of Q extends to an automorphism. Third is the fact that, if X, Y are finite ordered sets with X ç 7, then any embedding of X in Q can be extended to an embedding of Y. These observations are the starting point for the next section. For a fuller and more leisurely discussion of oligomorphic permutation groups, see my lecture notes (Cameron 1990). 2. Xo-Categoricity and Homogeneity A countable structure M over a first-order language is Wo-categorical if it is the unique countable model of its theory, i.e. determined up to isomorphism by countability and first-order sentences. (The prototype is Q, characterized by Cantor's theorem, as we saw.) In the spirit of geometry (where, since Klein's Erlanger Programm, we have known of a connection between axiomatizability and symmetry), the following remarkable result was found independently by Engeler (1959), Ryll-Nardzewski (1959) and Svenonius (1959): Theorem 2.1. The countable structure M is ^-categorical if and only if Aut(M) is oligomorphic. The proof involves the well-known tool of "back-and-forth". Without going into details (familiar to the experts), I note that back-and-forth is used, not just to show that two countable structures are isomorphic, but also to describe orbits on n-tuples structurally. The pre-requisite for back-and-forth is the ability to extend finite isomorphisms one point at a time. A countable relational structure M is homogeneous if every isomorphism between finite substructures of M can be extended to an automorphism of M. The age of M, written Age(M), is the class of finite structures embeddable in M. Now back-and-forth shows that M is homogeneous if and only if, for any X,Y e Age(M) with X Y, any embedding of X into M can be extended to Y. It suffices to require this when \Y\ = \X\ + 1. Using these ideas, Fraïssé (1953) showed: Theorem 2.2. A class sé of finite structures is the age of a countable homogeneous structure M if and only if sé is closed under isomorphism and under taking substructures, has only countably many non-isomorphic members, and has the amalgamation property. If these conditions hold, then M is unique up to isomorphism.

3 Infinite Permutation Groups in Enumeration and Model Theory 1433 We say that M is the Fraïssê limit of the class sé. This result is extremely useful for constructing examples. For instance, the class of finite triangle-free graphs has Fraïssé's properties; so there is a unique countable homogeneous universal triangle-free graph T (Henson 1971). There is a natural topology on the symmetric group SOD, that of pointwise convergence. In this topology, the full automorphism group of any first-order structure is closed in S^. Moreover, for G < H, G is a dense subgroup of H if and only if G and H have the same orbits on n-tuples for all n. Now, given any permutation group G, there is a "canonical relational structure" M such that G is a dense subgroup of Aut(M), and M is homogeneous. The automorphism group of a countable homogenous structure over a finite relational language is thus a closed oligomorphic permutation group. The converse is false, however; I do not know any nice characterization of this class of permutation groups. (The closed oligomorphic groups are precisely the automorphism groups of countable Xo-categorical structures; we may add the word "homogeneous" to the right-hand side of this equivalence.) If M is a homogeneous structure and G = Aut(M) is oligomorphic, then / (G) is equal to the number of unlabelled 77-element structures in the class Age(M) (i.e. up to isomorphism), and F n (G) to the number of labelled structures (i.e. on the point set {0,...,n 1}). Thus, enumeration of unlabelled or labelled structures in classes satisfying Fraïssé's hypotheses is equivalent to description of the appropriate orbit-counting sequence for an oligomorphic group G. As hinted above, many interesting combinatorial enumeration problems are of this type. 3. Generating Functions It is common practice in combinatorial enumeration to use different forms of generating function in labelled and unlabelled enumeration problems (Goulden and Jackson 1983). Thus, we define the ordinary generating function»2>0 for (/«(G), and the exponential generating function for (F n (G)). Convergence properties of these functions connect with growth rates (e.g. finite non-zero radius of convergence of fc is equivalent to exponential growth of (fn(g))), but usually we treat the series formally. Both turn out to be specializations of a series in infinitely many variables, as follows. If if is a finite permutation group of degree n, its cycle index is the polynomial Z(H; Sl s n ) = ~Vsf h K..^, where Cd(h) is the number of d-cycles in the cycle decomposition of h. Its rôle in enumeration is well known. Now, if G is a finite or oligomorphic permutation group, we define the modified cycle index of G to be

4 1434 Peter J. Cameron Z(G;si,s 2,...) = ^Z(Hì;S U S 2,...), where the summation is over representatives of the G-orbits on finite setsj and Hi is the group induced on the i-th set by its setwise stabilizer. (By convention, the empty set contributes a term 1.) For finite G, it can be shown that Z(G;s 1,S2,...) = Z(G;s 1 + l,s 2 + l,...). A couple of examples for infinite groups are and Z(S co ) = exp(-^ ) Z{A) = These series specialize as follows: i n>l 1 1-si Proposition 3.1. For any oligomorphic permutation group G, (a)f G (t) = Z(G;t,t 2,t 3,...); (b)f G (t)=z(g;t,0,0,...). The behaviour under direct and wreath products and point stabilizers can be described: Proposition 3.2. (a) Z(GxH) = Z(G)Z(H); (b) Z(GWrH) = Z{H;Z(G) - 1) ; (c),2(0,,,) = 2(G). (The substitution in (b) is defined by A(B) = A(B(s u 5 2, s 3,...), B(s 2, s*, s 6,...), B(s 3, s 6, s 9,...),...). In (c), the summation is over a set of orbit representatives 0^; G ai is the stabilizer of a,-, acting on the remaining points.) Thus, fgwrh can be determined from fc and Z(H). For example, fgmsjt) = Y[(l-t n )- fnig \ /GWrxW = ^ 7-7^. 2-/GW I close this section with a few simple examples. Example 1. fc 2 (t) = ; so /c 2 wr^(0 = 1/(1 t t 2 ). This is the generating function for the Fibonacci numbers. Example 2. f n (Soo WrSoo) is the partition function p(n), whose generating function is rin>i(l ~" t n )~ l. Moreover, the generating function for f n (Sao Wr S œ Wr S^) is Iln>i(l t n )~ p ( n \ which converges for all t; so the growth rate is slower than

5 Infinite Permutation Groups in Enumeration and Model Theory 1435 exponential, though faster than exp(n 1_E ) for any e > 0. this sequence arises in work of Cayley (1889) counting canonical forms. Example 3. We have F G wrh W = F/j(F G ( ) 1), where the usual substitution is intended. In particular, F 5œ WrG(0 = F G (é - 1). Let F*(t) be the exponential generating function for F*(G). Since F*(G) = F (SooWrG), we have F* G (t) = F G (e t - 1), which is thus equivalent to the relation involving Stirling numbers given in Section 1. Example 4. Let C be the group preserving the cyclic order on the roots of unity. Then n^l d\n From the fact that fc(t) = 1/(1 t), it can be deduced that exp(t/(l - t)) = Y[(l - f)~^{n)/n.»;>i For several further amusing examples, including groups realizing several familiar sequences, see Cameron (1987b), (1989). There are also close connections with Joyal's (1981) combinatorial formal power series. (Joyal regards an age as a category whose morphisms are the embeddings; the objects of cardinality n occur as the coefficients in formal power series.) 4. The Random Graph It follows from Fraïssé's theorem (2.2) that there is a unique countable homogeneous graph JR in which every finite graph is embedded. R is characterized by the property that, if U and V are finite disjoint sets of vertices, there is a vertex z joined to every vertex in U and to no vertex in V. (This property translates the equivalent of homogeneity given just before the statement of (2.2).) The graph R first appeared in the literature in a paper by Erdös and Rényi (1963), who showed that, with probability 1, a countable random graph is isomorphic to R. (The probability measure is defined by the rule "choose edges independently with probability ^"; but in fact the same graph is obtained if we take any fixed edge probability p with 0 < p < 1, or even if we let p vary a bit, e.g. tend to infinity not too slowly.) The proof of this paradoxical assertion is remarkably easy. First, given fixed finite disjoint sets U and V, the probability that the required vertex exists is 1. Now, since there are only countably many choices for U and V, and a countable intersection of sets of measure 1 has measure 1, the characteristic property of R holds with probability 1. Although an explicit description of JR is unnecessary for this d iscussion (the probabilistic argument shows its existence, while back-and-forth gives uniqueness), there are a couple of simple constructions for it:

6 1436 Peter J. Cameron Construction 1. The vertices are the natural numbers; x is joined to y if x < y and 2 X occurs in the (binary) expression for y as a sum of distinct powers of 2 (or vice versa). Construction 2. The vertices are the primes congruent to 1 (mod 4); p and q are joined if p is a quadratic residue mod q. (This is symmetric, by quadratic reciprocity.) In Construction 1, the asymmetric form of the relation gives a model for the Zermelo-Fraenkel axioms of set theory excluding the axiom of infinity. The proof of the characteristic property of R in Construction 2 is a pleasant exercise using the Chinese Remainder Theorem and Dirichlet's Theorem. As noted, the back-and-forth argument shows not only the uniqueness of R, but the homogeneous action of its automorphism group. If we use the first explicit construction of R, we find a group of primitive recursive automorphisms acting homogeneously on R. It would be interesting to investigate this group; for example, to see how the "recursive presentation" of R affects the structure of the group. Truss (1985) showed that the full automorphism group of R is simple, and described all the cycle types of its elements. We know that F n (Aut(R)) and f n (Aut(R)) are equal to the numbers of labelled and unlabelled graphs on n vertices respectively; the former is 22 n (" _1 \ the latter asymptotically 22 n ("~ 1 )/n!. R is not the only countable homogeneous graph. All such graphs were found by Lachlan and Woodrow (1980) (the finite ones had been found earlier by Gardiner (1976)). Theorem 4.1. A countably infinite homogeneous graph is one of the following: (i) the disjoint union of m complete graphs of size n, where at least one of m and n is infinite; (ii) complement of (i) (complete multipartite); (iii) the Fraïssê limit of the class of finite graphs containing no complete subgraph of size n (n> Ì); (iv) complement of (iii); (v) the random graph R. The universal K w -free graphs in (iii) were constructed by Henson (1971). We met Henson's graph for n = 3 earlier, where it was called T. I turn now to more general relational structures. Suppose that we have some class 2 of objects, each of which is described by a countable sequence of choices. There are two ways of assigning structure to the set!%: Method 1. By assigning non-negative numbers summing to 1 to the outcomes of each choice, becomes a measure space. For example, a countable graph can be determined by choosing "edge" or "non-edge" for each pair of vertices; if we give each choice the value \, we obtain the above model. Equivalently, we could assign 1/2" to each possible extension of an n-vertex graph to an (n + 1)-vertex graph. This technique can in principle be extended to many other classes of relational structures; but it is not clear what values to assign in general.

7 Infinite Permutation Groups in Enumeration and Model Theory 1437 Method 2. We can make SE into a complete metric space by defining the distance between two objects to be a suitable (decreasing) functions of the length of the common initial sequence of choices defining them. (The longer we have to wait to distinguish two objects, the closer they are.) Although the metric involves a choice of function, the topology does not. A set J ^ SE is open if, for any Yef, there is an initial subsequence of the choice sequence defining Y which forces membership in ^; and a set / ç SE is dense if, following any finite number of choices, there is a continuation defining a member of <&. A set is residual if it contains a countable intersection of open dense sets. The Baire category theorem asserts that, in a complete metric space, a residual set is non-empty (and, perforce, dense). Residual sets are regarded as "large", comparable to sets of measure 1 in a probability space. Analogously to the Erdos-Rényi theorem, it is possible to show that the set of graphs isomorphic to R is residual in the set of countable graphs. We now generalize this observation. Let sé be any age. We define SC(sé) to be the class of all structures on the set N whose age is contained in sé. (Thus, for example, if sé = Age(#), then S (sé) is the set of all graphs on the vertex set N.) An element of sé is determined by countably many choices, the w th choice describing how to extend an element of sé on the set {0,...,n 1} to one on the set {0,...,/?}. So SE (sé) is a complete metric space. Now we have: Proposition 4.2. Let M be a countable homogeneous structure. Then the set of structures isomorphic to M is residual in SE(Kgs(M)). No such result holds for measure, which seems much harder to deal with. However, there are some specific results. For example, Q is the random total order (where the measure is defined by making all possible orderings of any finite set equally likely). 5. Growth Rates Quite a bit is known about possible growth rates of the sequence (f (G)). (Of course, these general results apply to the numbers of unlabelled structures in a class satisfying Fraïssé's conditions. However, some of them are known to hold for any age.) A basic fact is that this sequence is non-decreasing: see Cameron (1976). Most of the results are due to Pouzet (1981) and Macpherson (1985a), (1985b). Pouzet showed that the rate of growth is either polynomial (an d < f u (G) < bn d, where n e N and a,b > 0) or faster than any polynomial. In the latter case, Macpherson found a fractional exponential lower bound exp(m~ E ), comparable to the partition function. For primitive groups, Macpherson's result is much more striking: Theorem 5.1. There is an absolute constant c > 1 such that, if G is primitive, then either f n (G) = 1 for all n, or f n (G) > c" for all sufficiently large n.

8 1438 Peter J. Cameron Macpherson gave c = 2$ s; it is conjectured that the result holds with c = 2 e (this would be best possible, see below). Polynomial growth of degree fc 1 is realized by, among others, S (acting with k orbits) and S^WrSk (acting imprimitively). S«) WrS«, realizes the partition function. Other fractional exponential growth rates, roughly exp(«p+ 2 ) for p G N, can also be realized. We saw that the growth rate for S œ WrSoo WrSoo is faster than fractional exponential but slower than exponential. There are many imprimitive examples with exponential growth; we saw the Fibonacci numbers realized by C 2 WrA Primitive groups exhibiting exponential growth are fairly rare. Most of them are automorphism groups of "treelike objects" (Cameron 1987b) related to the Q-trees of combinatorial group theory (Alperin and Bass 1987). There are also structures related to circular orders, including Lachlan's (1984) "circular tournament" L. The group of order preserving and reversing permutations of L has the slowest known growth rate of any primitive group (/ (Aut(L)) ~ 2 n ~ 2 /n). For growth rates faster than exponential, we see in nature a gap between factorial growth (for the homogeneous pair of linear orders, we have f n (G) = nl), and growth like exp(cn 2 ) (realized by the random graph, and projective and affine spaces over finite fields). A result of Macpherson (1987) throws some light on this. The independence property (Shelah 1978) forces growth at least exp(cn 2 ); for homogeneous structures over finite languages, negating the independence property bounds the growth by exp(n 1+e ). Also, for co-stable structures, the same gap occurs, the criterion being the types of strictly minimal sets around which the structure is built. (See Cherlin-Harrington-Lachlan (1985) for the theory of co-stable, No-categorical structures.) In general, there is no upper bound for the rate of growth of (f n (G)): take a homogeneous structure over a language where the number of n-ary relation symbols grows as fast as you please with n. On the other hand, for a homogeneous structure over a finite language, f n (G) < exp(p(n)) for some polynomial P. Many of the most interesting open questions about growth rate concern its smoothness. For example, do the limits \im n^o0 f n (G)/n d lim^oo log log f n (G)/ log n lim w _ >00 / (G) 1>/n (for polynomial growth of degree d), (for fractional exponential growth), (for exponential growth), exist? If so, what possible values can these limits take? Apart from Macpherson's gap in values of the third limit for primitive groups, almost nothing is known. It is known that f n (G) = f n +i(g) = f n +i(g) can only hold in the trivial case where G fixes a set of size at most n and is transitive on (n -{- 2)-subsets of the complement. The situation f n (G) = f n+ \(G) has been studied; a few examples are known, and some have been characterized, but a general result seems difficult.

9 Infinite Permutation Groups in Enumeration and Model Theory Cyclic Automorphisms of Graphs Measure and Baire category have been used for constructing subgroups of oligomorphic groups. Though much more general results are available, I will consider here only a special case, regular cyclic subgroups. Let g be a cyclic automorphism of a countable graph J\ Then the vertices of r can be indexed by the integers in such a way that g acts as a shift d\ \-* a, + i. Let S = S(r,g) = {n > 0 : ao ~ a,,}. Then S determines F up to isomorphism (a/ ~ ocj if and only if / j\ e S), and g up to conjugacy in Aut(F). We write r = rs, g = gs. Now we can ask: for which sets S is r$ isomorphic to some interesting graph? A subset S of N is determined in an obvious way by infinitely many choices; so the methods of measure and Baire category apply. S is called universal if every finite sequence of zeros and ones occurs as a consecutive subsequence of the characteristic function of S. It is easily checked that rs = R if and only if S is universal. A weak form of the law of large numbers says that the set of universal sequences has measure 1 ; it can also be shown to be residual. Since residual sets and measure-1 sets have cardinality 2 K, we conclude that the random graph has 2 Ko non-conjugate cyclic automorphisms! Now consider the homogeneous universal triangle-free graph T. First note that, for S N, Ts is triangle-free if and only if S is sum-free, i.e. x,y e S => x + y S. Now sum-free sets can be determined by countably many binary choices, in an obvious way: considering natural numbers in turn, if n = x + y where x,y have already been put into S, then n ^ S; otherwise we are free to choose. Let S be a sum-free set. The only obvious necessary condition for a finite zero-one sequence c to be a subsequence of the characteristic function of S is that, if j i G S, then e/ and sj cannot both be 1. Call a sum-free set sf-universal if every finite zero-one sequence satisfying this condition is a subsequence of the characteristic function of S. Now Fs = T if and only if S is sf-universal. Henson (1971) showed that T has cyclic automorphisms, i.e. that sf-universal sets exist. Can we prove this using category or measure? It is easily seen that a residual subset of all sum-free sets are sf-universal, so we do indeed get 2 Ko non-conjugate cyclic automorphisms of T. (Incidentally, I conjecture that an sfuniversal set has density 0. This would, if true, give a "density version" of Schur's theorem (1916) that N cannot be covered by finitely many sum-free sets.) However, when we turn to measure, the situation is different. Any set of odd numbers is obviously sum-free, and no such set is sf-universal. It came as a surprise to me to find that the probability that a random sum-free set consists entirely of odd numbers is non-zero. (This probability is about ) Furthermore, there are infinitely many "periodic" sum-free sets whose subsets have positive probability. (After the odd numbers, the next two are the congruence classes 2 and 3 (mod 5) and the congruence classes 1 and 4 (mod 5).) I do not know whether or not almost all sum-free sets are contained in periodic ones. (For details, see Cameron (1987a).) Experiment suggests the possibility of quasi-periodic behaviour : a periodicity is established for a few cycles and then disrupted, to return with its phase shifted as part of a longer period. Maybe this process can continue infinitely and yield non-periodic sets whose subsets have positive probability.

10 1440 Peter J. Cameron What of the corresponding graphs F^? For each periodic set S whose subsets have positive probability, there is an "almost homogeneous" graph F * such that r s > = T* for almost all subsets S f of S. (For the set of odd numbers, F* is the "universal bipartite graph".) It is not known whether any other triangle-free graphs occur with positive probability. (In particular, this is not known for T.) Many other questions about random sum-free sets remain open. For example, what is the average density, and how is the density distributed? (There are "spectral lines" corresponding to the periodic sets described above, e.g. a deltafunction of weight at density \ (from the sets of odd numbers). Does almost every sum-free set have a density? Is the density almost surely in the interval (0, ]? Does the spectrum have a continuous part?) I conclude with an example with very different behaviour. Covington (1989) calls a graph N-free if it contains no induced path of length 3. She showed that there is an "almost homogeneous" universal countable N-free graph C, unique up to isomorphism, but admitting no cyclic automorphisms. One can write down a condition on sets S equivalent to Ts being N-free. There are 2^ sets satisfying this condition, but the corresponding graphs rs are pairwise non-isomorphic! In other words, an N-free graph has at most one conjugacy class of cyclic automorphisms. References Alperin, R., Bass, H. (1987): Length functions of group actions on ^1-trees. In: Combinatorial group theory and topology (Alta, Utah, 1984) (Ann. Math. Stud. 111). Princeton Univ. Press, Princeton, NJ, pp Cameron, P.J. (1976) : Transitivity of permutation groups on unordered sets. Math. Z. 148, Cameron, P.J. (1987a) : Portrait of a typical sum-free set. In: Whitehead, C. (ed), Surveys in combinatorics. London Math. Soc. (Lecture Notes, vol. 123). Cambridge Univ. Press, Cambridge, pp Cameron, P.J. (1987b): Some treelike objects. Quart. J. Math. Oxford (2) 38, Cameron, P.J. (1989) : Some sequences of integers. Discrete Math. 75, Cameron, P.J. (1990): Oligomorphic Permutation Groups. London Math. Soc. (Lecture Notes, vol. 152). Cambridge Univ. Press, Cambridge Cameron, P.J., Taylor, D.E. (1985) : Stirling numbers and affine equivalence. Ars Combinatoria 20B, 2-14 Cantor, G. (1895): Beiträge zur Begründung der transfiniten Menge. Math. Ann. 46, Cayley, A. (1889) : Collected mathematical papers. Cambridge Univ. Press, London Cherlin, G.L., Harrington, LA., Lachlan, A.H. (1985) : Ko-categorical, X 0 -stable structures. Ann. Pure Appi. Logic 28, Covington, J. (1989): A universal structure for JV-free graphs. Proc. London Math. Soc. (3) 58, 1-16 Engeler, E. (1959) : Äquivalenzklassen von n-tupeln. Z. Math. Logik Grundl. Math. 5, Erdös, P., Rényi, A. (1963): Asymmetrie graphs. Acta Math. Acad. Sei. Hungar. 14, Fraïssé, R. (1953) : Sur certains relations qui généralisent l'ordre des nombres rationnels. C. R. Acad. Sci. Paris 237, Gardiner, A.D. (1976) : Homogeneous graphs. J. Comb. Theory (B) 20,

11 Infinite Permutation Groups in Enumeration and Model Theory 1441 Goulden, LP., Jackson, D.M. (1983): Combinatorial enumeration. Wiley, New York Henson, C.W. (1971) : A family of countable homogeneous graphs. Pacific J. Math. 38, Joyal, A. (1981): Une théorie combinatoire des séries formelles. Adv. Math. 42, 1-82 Lachlan, A.H. (1984): Countable homogeneous tournaments. Trans. Amer. Math. Soc. 284, Lachlan, A.H., Woodrow, R.E. (1980): Countable ultrahomogeneous undirected graphs. Trans. Amer. Math. Soc. 262, Macpherson, H.D. (1985a) : Orbits of infinite permutation groups. Proc. London Math. Soc. (3) 51, Macpherson, H.D. (1985b) : Growth rates in infinite graphs and permutation groups. Proc. London Math. Soc. (3) 51, Macpherson, H.D. (1987) : Permutation groups of rapid growth. J. London Math. Soc. (2) 35, Pouzet, M. (1981): Application de la notion de relation presque-enchaïnable au dénombrement des restrictionsfinisd'une relation. Z. Math. Logik Grundl. Math. 27, Ryll-Nardzewski, C. (1959): On category in power < Ko. Bull. Acad. Poi. Sér. Math. Astr. Phys. 7, Schur, I. (1916): Über die Kongruenz x m + y m = z'"(modp). Jber. Deutsch. Math.-Verein. 25, Shelah, S. (1978) : Classification theory and the number of non-isomorphic models. North- Holland, Amsterdam Svenonius, L. (1959) : Ko-categoricity in first-order predicate calculus. Theoria 25, Truss, J.K. (1985) : The group of the countable universal graph. Math. Proc. Camb. Philos. Soc. 98,

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