Length of an intersection

A poset \bfp is well-partially ordered (WPO) if all its linear extensions are well orders~; the supremum of ordered types of these linear extensions is the {\em length}, \ell(\bfp) of \bfp. We prove that if the vertex set $X$ of \bfp is infinite, of cardinality $\kappa$, and the ordering $\leq$ is the intersection of finitely many partial orderings $\leq_i$ on $X$, $1\leq i\leq n$, then, letting \ell(X,\leq_i)=\kappa\multordby q_i+r_i, with $r_i<\kappa$, denote the euclidian division by $\kappa$ (seen as an initial ordinal) of the length of the corresponding poset~: \ell(\bfp)< \kappa\multordby\bigotimes_{1\leq i\leq n}q_i+ \Big|\sum_{1\leq i\leq n} r_i\Big|^+ where $|\sum r_i|^+$ denotes the least initial ordinal greater than the ordinal $\sum r_i$. This inequality is optimal (for $n\geq 2$).Comment: 13 page

Indivisible ultrametric spaces

A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces, we show that a countable ultrametric space embeds isometrically into an indivisible ultrametric metric space if and only if it does not contain a strictly increasing sequence of balls.Comment: 21 page

Atomic compactness for reflexive graphs

A first order structure $\mathfrak{M}$ with universe M is atomic compact if every system of atomic formulas with parameters in M is satisfiable in $\mathfrak{M}$ provided each of its finite subsystems is. We consider atomic compactness for the class of reflexive (symmetric) graphs. In particular, we investigate the extent to which "sparse" graphs (i.e. graphs with "few" vertices of "high" degree) are compact with respect to systems of atomic formulas with "few" unknowns, on the one hand, and are pure restrictions of their Stone-Čech compactifications, on the other hand

Orthogonal Countable Linear Orders

International audienceTwo linear orderings of a same set are perpendicular if every self-mapping of this set that preserves them both is constant or the identity. Two isomorphy types of linear orderings are orthogonal if there exist two perpendicular orderings of these types. Our main result is a characterisation of orthogonality to ω : a countably infinite type is orthogonal toω if and only if it is scattered and does not admit any embedding into the chain of infinite classes of its Hausdorff congruence. Besides we prove that a countable type is orthogonal toω + n (2 ≤ n < ω) if and only if it has infinitely many vertices that are isolated for the order topology. We also prove that a typeτ is orthogonal to ω + 1 if and only if it has a decomposition of the formτ = τ1 + 1 + τ2 withτ1 orτ2 orthogonal to ω, or one of them finite nonempty and the other one orthogonal toω + 2. Since it was previously known that two countable types are orthogonal whenever each one has two disjoint infinite intervals, this completes a characterisation of orthogonality of pairs of types of countable linear orderings. It follows that the equivalence relation of indistinguishability for the orthogonality relation on the class of countably infinite linear orders has exactly seven classes : the classes respectively of ω, ω + 1, ω + 2, ω + ω, ω ω , 3 ⋅ η and η, where η is the type of the ordering of rational numbers and 3 ⋅ η is the lexicographical sum along η of three element linear orders

Countable linear orders with disjoint infinite intervals are mutually orthogonal

International audienceTwo linear orderings of a same set are perpendicular if the only self-mappings of this set that preserve them both are the identity and the constant mappings. Two linear orderings are orthogonal if they are isomorphic to two perpendicular linear orderings. We show that two countable linear orderings are orthogonal as soon as each one has two disjoint infinite intervals. From this and previously known results it follows in particular that each countably infinite linear ordering is orthogonal to itsel

The length of an intersection

International audienc