A Pigeonhole Property for Relational Structures

Transcription

1 ~~~ ~~ ~ ~~~ Math. Log. Quart. 45 (1999) 3, Mathematical Logic WILEY-VCH Verlag Berlin GmbH 1999 A Pigeonhole Property for Relational Structures Anthony Bonatoa and Dejan Deli6 ') a Department of Mathematics and Computer Science, Mount Allison University 72 York St., Sackville NB, E4L 1E8 Canada2) Department of Pure Mathematics, University of Waterloo, Waterloo ON, N2L 3G1 Canada3) Abstract. We study those relational structures S with the property (P) that each partition of S contains a block isomorphic to S. We show that the Fraissh limits of parametric classes ic have property (P); over a binary language, every countable structure in Ic' satisfying (P) along with a condition on 1-extensions must be isomorphic to this limit. Mathematics Subject Classification: 03C15, 03C50. Keywords: Pigeonhole property, FrafsssSi limits, parametric classes 1 Introduction As proved first by HENSON in [4], the countable random graph R (also called the countable universal homogeneous graph) has the following unusual property: if the vertices of R are partitioned into finitely many sets XI,...,X,, then for some 1 5 i 5 n the induced subgraph on Xi is isomorphic to R. CAMERON (see [l] and [2]) has called this property the pigeonhole property. One can generalize the pigeonhole property from graphs to relational structures in the following way: D e f i nit ion 1.1. Let L be a relational language and let S be an L-structure. The structure S has property (P) if IS1 > 1 and for each n 2 2, whenever S = S1 u'..us,,, then for some 1 5 i 5 n, S 1 Si Z S, where kj is disjoint union, and S Si is the induced substructure on S, in S. CAMERON asked the following question: what "nice" structures have property (P)? (see [l, p. 481). We believe that Theorem 1.1 below gives a satisfactory answer to this question within many classes of relational structures. Before we can state Theorem 1.1, we must first supply some definitions. For background in basic model theory, the reader is referred to either [3] or [5]; see [5] for results on FraissC limits and e. c. structures. For L-structures S, TI we write S 5 T if S is a substructure of T. Throughout, we assume L is a relational language. ')The second author was supported by an 0. G. S. scholarship. 2) abonato@lmta.ca 3) ddelic@lbarrow.uwaterloo.ca

2 410 Anthony Bonato and Dejan Deli6 Definition 1.2. (1) Let K be a class of L-structures closed under isomorphisms. A structure S E K is l-e. c. iff for all T E ic, a E S and 0(z, a) a quantifier-free L-formula, if S 5 T and T k 32 O(zJ a), then S I= 32 O(z, a). (2) For a first-order L-formula 8, V (distinct)zl... z,8 abbreviates Vzl... zn(/\lsi<j<,, xi # zj + 0). (3) K is a parametric class over L if the following conditions hold: (a) ic is axiomatized by a finite set of L-sentences of the form V (distinct)xl... x,o, where 0 is a Boolean combination of atomic L-formulas of the form Ryl... ym with { Y1 J..., Ym} = {z1,...,xn} ; (b) K contains infinite structures; (c) there is only one isomorphism type of one element structure in K. Theorem 1.1. Let X: be a parametric class over a finite relational language L without unary predicate symbols. (1) K has a Frai'sse' limit F(K) and F(K) has property (P). (2) If L consists of only binary predicate symbols, then for a countable S E K the following are equivalent: (a) S satisfies property (P) and S is 1-e. c. (b) S S F(K). Remark 1.1. (1) Parametric classes were first studied by W. OBERSCHELP in [6] in the context of 0-1 laws in finite model theory. See also [3, Chap. 31 for further details. (2) The hypothesis in Theorem 1.1 that L contain no unary predicate symbols is essential; further, the restriction on l-element structures in parametric classes cannot be dropped. See Lemma 2.3 below. (3) We do not know of an extension of Theorem 1.1(2) to languages with predicate symbols of arity Examples We defer the proof of Theorem 1.1 to the end of the article and first consider some of its consequences. 2.1 Applications of Theorem The class of k-uniform hypergraphs is axiomatized by the following parametric sentences: V (distinct)zl... Xk (R(q... zk) 3 (R(Z,(~)... z+))), where sk is the symmetric group of order k, and for each 15 j 5 k - 1, V(distinct)~l...Xj,_.., rj+{yl,..., yhl -R(Y~...~k)). By Theorem 1.1( 1) the universal homogeneous &uniform hypergraph has property (P).

3 A Pigeonhole Property for Relational Structures The following classes are parametric over a language with a single binary predicate symbol {E}; hence by Theorem 1.1(2) the countable universal homogeneous structure in each class is characterized by having property (P) and by being 1-e. c. In each case, we give a parametric axiomatization: (a) Digraphs: Vx (7xEz). (b) Graphs: Vx (ixex), V(distinct)xy(zEy + yec). (c) Oriented (or asymmetric) graphs: Vx (ixex), V (distinct)zy (xey -, 7yEx). (d) Tournaments: Vx (izez), V(distinct)xy (cey c) 7yEx). (e) Undirected tournaments: Vx (~zex), V (distinct)zy (xey V yez) Structures without property (P) Definition 2.1. Let S be an L-structure. Define the graph of S, written G(S), to be the graph with vertices S and edges ((2, y) : x, y E S so that x # y and there exists R E L of arity n 2 2 with {al,...,a,} S so that 2, y E {al,...,a,} and R al...a,}. Lemma 2.1. Let L consist of only binary predicate symbols, and let S be an L-structure satisfying property (P). Then G(S) satisfies property (P). Proof. Let G(S) = S1 kj...us,. Then S = S1 kj...us,, so for some 15 i 5 n, S 1 Si S. Thus, G(S 1 Si)E G(S). But G(S 1 Si) = G(S) 1 Si. cl Corollary 2.2. Let L be as in Lemma 2.1. Let s be a countable L-structure so that G(S) is not K or KN~. If G(S) does not have every countable graph as an induced subgraph, then G(S) does not have property (P). In particular, the universal homogeneous order P does not satisfy property (P). Proof. In [2], CAMERON proved that the only countable graphs with property (P) are KN,,, K, and R. The first statement now follows by Lemma 2.1, as G(S) must be R, and R embeds every countable graph as an induced subgraph. For the final statement of the corollary, note that G(P) is just the comparability graph of P, and so has no induced subgraph isomorphic to the five cycle. 0 Lemma 2.3. (1) Let S be a structure with more than one isomorphism type of one element structure. Then S does not have property (P). (2) FOT every finite nonempty language L, the random L-structure (the Fraasse limit of the class of finite L-structures) does not have property (P). (3) If L contains unary predicate symbols and S interprets these predicate symbols nontrivially (that is, for some P E L, 8 # Ps # S), then S does not have prop- erty (P), Proof. The proofs of (1) and (3) are left to the reader. For (2), note that the random L-structure contains more than one isomorphism type of one element structure. For example, by universality, there is a one element substructure with empty relations, and for R E L there is a one element structure S with domain {a} whose only relation is RSa... a. 0

4 412 Anthony Bonato and Dejan Deli6 3 Proof of Theorem 1.1 (1) The proof of the existence of F(K) in K is implicit in the discussion of [3, Section 3.21, and so we omit the details. We now show that F(K) has property (P). Fix n 2 2. Let F(K) = S1 M...M S,. If F(K) does not have property (P), then there are finite sets Xi 5 Si so that F(K) Xi has a one element extension Y; with domain Xi U {d;} not realized in F(K) [ Si (we are using the fact that F(K) is the unique countable e. c. model of K). Let A = F(K) 1 U:=l Xi. By hypothesis, we may identify the Yi [ {dj} s; let the isomorphism type of Y; 1 {d;} have domain {d}. Define a structure B with domain A U {d} and with relations those of A, the relations in Yi induced by the extension by d of each Xi and the set of relations defined as follows: By hypothesis, there is a K-structure C of infinite cardinality. Let m = max(arity (R): R E L) and fix a tuple E of length m from C. For each tuple ii from B of length 5 m, so that d occurs in u and for each 1 I. i 5 n, 7i does not list the elements of Xi U {d}, let C be a sub-tuple of C of length la[, and for each R E L, a E RB if and only if i? f RC. With the definition of the relations of B just given, each tuple of distinct elements of appropriate length from B will satisfy the axioms of K, so B E K. Hence, B is an (/A/ + 1)-element K-extension of A 5 F(K), and so may be realized in F(K). If d is realized in F(K) 1 S,, for some 1 5 i 5 n, then by construction Y; is realized in F(K) S,, contrary to hypothesis. (2) By (I), F(K) satisfies property (P). The limit F(K) is 1-e. c. as it is e. c. For the converse, we show that S is e.c. The proof is by induction on n 2 1. The induction hypothesis is as follows: If S 5 T, T E K and for 6 E S, 6(z,u) is a quantifier-free L-formula, then T k 32 6(z, ii) implies S I= 3z e(z, 5) (note that we only require a single variable in 3z O(z, ii); the general case of a formula of the form 3% O(%, a) with 5 a finite tuple of variables will follow by induction). The case n = 1 holds as S is 1-e.c. Let S 5 T and T E K so that for an (n + 1)-element subset {al,..., a,+l} E S and for a quantifier-free L-formula 6(z, a],...,a,+l), T k 3zB(2, u1,..., a,+l). Without loss of generality, we can as- sume that 6(z, a],..., a,+l) has the form 6 (z, a,+l) A V(z, al,..., a,) with suitable quantifier-free L-formulas 6 (z, an+l) (z, al,..., a,) (since L contains only binary symbols and a subformula containing no z is automatically true in S). Furthermore, we can assume that each witness to 326(z, al,..., a,+l) in T is distinct from each element of {al,..,, a,+l} (otherwise, there is a witness to 3r6(z, al,..., a,+l) in S). In particular, we may assume that 6(z, a1,..., a,+l) contains the conjunc- tion A15i5,+l z # ai, 6 (z, a,+]) contains z # an+lr and W(2, al,...,a,) contains AIsis,, z # a,. Define the following subsets of S: R = {C E S : S Cr B(c, ~ 1,.., ~n+l)}, R = {c E S : S I= 6 (c, un+l)}j Q = {C E S : S k B (c, all...,a,)}. We show that R # 8. By inductive hypothesis, both R # 8 and R # 8. Note that R = $2 n a. By hypothesis, a,+l $4 R and {a],...,a,} n R = 0. Let A = {a,+l} U R and let B = S - A. If S [ B Y S, as {a1,..., a,} C B, S 1 B F 3zB (z, a],..., a,,), as all the witnesses to 3z6 (z, all...,a,) in S are in A. But R # 8 in B (as a # 0 in S by inductive hypothesis). Contradiction.

5 A Pigeonhole Property for Relational Structures 413 Hence, as S satisfies property (P), S A S. It follows by inductive hypothesis that 52; = {c E A : S 1 A b O (c,a,+l)} # 0 so that 52; n 52 # 0. But then R/ n 52// # 0. O References [l] CAMERON, P. J., Oligomorphic Permutation Groups. London Math. SOC. Lecture Notes 152, Cambridge University Press, Cambridge [2] CAMERON, P. J., The random graph. Algorithms and Combinatorics 14 (1997), [3] EBBINGHAUS, H.-D., and J. FLUM, Finite Model Theory. Springer-Verlag, Berlin-Heidelberg-New York [4] HENSON, C. W., A family of countable homogeneous graphs. Pacific J. Math. 38 (1971), [5] HODGES, W., Model Theory. Encyclopedia of Mathematics and its Applications, Vol. 42, Cambridge University Press, Cambridge [6] OBERSCHELP, W., Asymptotic 0-1 laws in combinatorics. Lecture Notes in Mathematics 969, Springer-Verlag, Berlin-Heidelberg-New York 1982, pp (Received: March 21, 1998)