Search

A metric space M is homogeneous if every isometry between finite subsets extends to a surjective isometry defined on the whole space. We show that if M is an ultrametric space, it suffices that isometries defined on singletons extend, i.e that the group of isometries of M acts transitively. We derive this fact from a result expressing that the arity of the group of isometries of an ultrametric space is at most 2. An illustration of this result with the notion of spectral homogeneity is given. With this, we show that the Cauchy completion of a homogeneous ultrametric space is homogeneous. We present several constructions of homogeneous ultrametric spaces, particularly the countable homogeneous ultrametric space, universal for rational distances, and its Cauchy completion. From a general embeddability result, we prove that every ultrametric space is embeddable into a homogeneous ultrametric space with the same set of distances values and we also derive three embeddability results due respectively to F. Delon, A. Lemin and V. Lemin, and V. Fevinberg. Looking at ultrametric spaces as 2-structures, we observe that the nerve of an ultrametric space is the tree of its robust modules.

We investigate infinite versions of vector and affine space partition results, and thus obtain examples and a counterexample for a partition problem for relational structures. In particular we provide two (related) examples of an age indivisible relational structure which is not weakly indivisible., Comment: Rev 1 (October 6, 2009)

We study the validity of a partition property known as weak indivisibility for the integer and the rational Urysohn metric spaces. We also compare weak indivisiblity to another partition property, called age-indivisibility, and provide an example of a countable ultrahomogeneous metric space which may be age-indivisible but not weakly indivisible., Comment: 23 pages; updated version made self-contained, last section modified

The distinguishing number of a graph $G$ is the smallest positive integer $r$ such that $G$ has a labeling of its vertices with $r$ labels for which there is no non-trivial automorphism of $G$ preserving these labels. Albertson and Collins computed the distinguishing number for various finite graphs, and Imrich, Klav\v{z}ar and Trofimov computed the distinguishing number of some infinite graphs, showing in particular that the Random Graph has distinguishing number 2. We compute the distinguishing number of various other finite and countable homogeneous structures, including undirected and directed graphs, and posets. We show that this number is in most cases two or infinite, and besides a few exceptions conjecture that this is so for all primitive homogeneous countable structures., Comment: 21 pages

We study the finite dimensional partition properties of the countable homogeneous dense local order. Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of Milliken's theorem on trees., Comment: 17 pages, 9 figures. This is the very last version of the paper. Several proofs were simplified and pictures were added

We solve the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for the Hilbert space in the context of the Urysohn universal metric space. This is achieved by solving a purely combinatorial problem involving a family of countable homogeneous metric spaces with finitely many distances., Comment: 21 pages, 1 figure. This is the post-last version of the paper. A mistake had to be fixed in the article by Lopez-Abad and Nguyen Van Th\'e quoted in appendix. Modifications were done accordingly

The \textit{age} of a relational structure $\mathfrak A$ of signature $\mu$ is the set $age(\mathfrak A)$ of its finite induced substructures, considered up to isomorphism. This is an ideal in the poset $\Omega_\mu$ consisting of finite structures of signature $\mu$ and ordered by embeddability. If the structures are made of infinitely many relations and if, among those, infinitely many are at least binary then there are ideals which do not come from an age. We provide many examples. We particularly look at metric spaces and offer several problems. We also provide an example of an ideal $I$ of isomorphism types of at most countable structures whose signature consists of a single ternary relation symbol. This ideal does not come from the set $\age_{\mathfrak I}(\mathfrak A)$ of isomorphism types of substructures of $\mathfrak A$ induced on the members of an ideal $\mathfrak I$ of sets. This answers a question due to R. Cusin and J.F. Pabion (1970)., Comment: 18 pages