A Pigeonhole Property for Relational Structures

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 proper t y . One can generalize the pigeonhole property from graphs to relational structures in the following way: D e f i n i t i o n 1.1. Let L be a relational language and let S be an L-structure. The structure S has proper t y ( 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.