The MSO+U theory of (N, <) is undecidable

We consider the logic MSO+U, which is monadic second-order logic extended with the unbounding quantifier. The unbounding quantifier is used to say that a property of finite sets holds for sets of arbitrarily large size. We prove that the logic is undecidable on infinite words, i.e. the MSO+U theory of (N,<) is undecidable. This settles an open problem about the logic, and improves a previous undecidability result, which used infinite trees and additional axioms from set theory.Comment: 9 pages, with 2 figure

Isometry groups among topological groups

It is shown that a topological group G is topologically isomorphic to the isometry group of a (complete) metric space iff G coincides with its G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete). It is also shown that for every Polish (resp. compact Polish; locally compact Polish) group G there is a complete (resp. proper) metric d on X inducing the topology of X such that G is isomorphic to Iso(X,d) where X = l_2 (resp. X = Q; X = Q\{point} where Q is the Hilbert cube). It is demonstrated that there are a separable Banach space E and a nonzero vector e in E such that G is isomorphic to the group of all (linear) isometries of E which leave the point e fixed. Similar results are proved for an arbitrary complete topological group.Comment: 30 page

A topological characterization of LF-spaces

We present a topological characterization of LF-spaces and detect small box-products that are (locally) homeomorphic to LF-spaces.Comment: 16 page

Stable graphs of bounded twin-width

We prove that every class of graphs $\mathscr C$ that is monadically stable and has bounded twin-width can be transduced from some class with bounded sparse twin-width. This generalizes analogous results for classes of bounded linear cliquewidth and of bounded cliquewidth. It also implies that monadically stable classes of bounded twin-widthare linearly $\chi$-bounded.Comment: 44 pages, 2 figure

Detecting topological groups which are (locally) homeomorphic to LF-spaces

We present a simple-to-apply criterion for recognizing topological groups that are (locally) homeomorphic to LF-spaces.Comment: 10 page

Indiscernibles and Flatness in Monadically Stable and Monadically NIP Classes

Monadically stable and monadically NIP classes of structures were initially studied in the context of model theory and defined in logical terms. They have recently attracted attention in the area of structural graph theory, as they generalize notions such as nowhere denseness, bounded cliquewidth, and bounded twinwidth. Our main result is the - to the best of our knowledge first - purely combinatorial characterization of monadically stable classes of graphs, in terms of a property dubbed flip-flatness. A class $\mathcal{C}$ of graphs is flip-flat if for every fixed radius $r$, every sufficiently large set of vertices of a graph $G \in \mathcal{C}$ contains a large subset of vertices with mutual distance larger than $r$, where the distance is measured in some graph $G'$ that can be obtained from $G$ by performing a bounded number of flips that swap edges and non-edges within a subset of vertices. Flip-flatness generalizes the notion of uniform quasi-wideness, which characterizes nowhere dense classes and had a key impact on the combinatorial and algorithmic treatment of nowhere dense classes. To obtain this result, we develop tools that also apply to the more general monadically NIP classes, based on the notion of indiscernible sequences from model theory. We show that in monadically stable and monadically NIP classes indiscernible sequences impose a strong combinatorial structure on their definable neighborhoods. All our proofs are constructive and yield efficient algorithms.Comment: v2: revised presentation; renamed flip-wideness to flip-flatness; changed the title from "Indiscernibles and Wideness [...]" to "Indiscernibles and Flatness [...]