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摘要
A relational structure H is homogeneous if every isomorphism between finite induced substructures has an extension to an automorphism of H. A relational structure R is indivisible if for every partition (A, B) of the set of elements of R, there is a copy of R in A or in B. The structure R is weakly indivisible if for every partition (A, B) of the set of elements of R for which some finite induced substructure of R does not have a copy in A, there exists a copy of R in B. The structure R is age indivisible if for every partition (A, B) of the elements of R every finite induced substructure of R has an embedding into A or every finite induced substructure of R has an embedding into B. It follows that indivisibility implies weak indivisibility which in turn implies age indivisibility. There are many examples of countable infinite homogeneous structures which are weakly indivisible but not indivisible; see Sauer (2000) [17] and the end of Section 3.1 of this paper. In general, it is difficult to find structures which are age indivisible but not weakly indivisible. One such example (see Pouzet et al. (2011) [16]) is obtained via the inhomogeneous general linear group of vector spaces over finite fields. Finding such an example for homogeneous structures seems to be even more difficult. The only one that I obtained, together with L Nguyen Van The, is a countable homogeneous metric subspace of the Hilbert sphere l(2) (see Nguyen Van The and Sauer (2010) [13]). Both of the examples above are relational structures with infinitely many relations. One of the still open questions then is: Are there countable infinite homogeneous structures with finite signature which are age but not weakly indivisible? We will show that the generic "free amalgamation" homogeneous structures are weakly indivisible; see Section 3.1. Countable "Urysohn metric spaces" (see the next section), with finite distance sets, are indivisible (see Sauer (2013) [19]). Countable Urysohn metric spaces are age indivisible, which follows from the Hales-Jewett theorem (Hales and Jewett (1963) [7]; see Delhomme et al. (2007) [2]). For non-Urysohn homogeneous countable metric spaces, except for the example mentioned above, the weak indivisibility question is completely open. We will show, in this paper, that a countable homogeneous ultrametric space is age indivisible if and only if it is weakly indivisible.
- 出版日期2014-4
