Decidability and Complexity of Tree Share Formulas

Fractional share models are used to reason about how multiple actors share ownership of resources. We examine the decidability and complexity of reasoning over the "tree share" model of Dockins et al. using first-order logic, or fragments thereof. We pinpoint a connection between the basic operations on trees union, intersection, and complement and countable atomless Boolean algebras, allowing us to obtain decidability with the precise complexity of both first-order and existential theories over the tree share model with the aforementioned operations. We establish a connection between the multiplication operation on trees and the theory of word equations, allowing us to derive the decidability of its existential theory and the undecidability of its full first-order theory. We prove that the full first-order theory over the model with both the Boolean operations and the restricted multiplication operation (with constants on the right hand side) is decidable via an embedding to tree-automatic structures

The approximate Loebl-Komlos-Sos conjecture and embedding trees in sparse graphs

Loebl, Koml\'os and S\'os conjectured that every $n$-vertex graph $G$ with at least $n/2$ vertices of degree at least $k$ contains each tree $T$ of order $k+1$ as a subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of $k$. For our proof, we use a structural decomposition which can be seen as an analogue of Szemer\'edi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of $G$ to embed a given tree $T$. The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050]

A proof of the rooted tree alternative conjecture

Bonato and Tardif conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or infinite. We prove the analogue of their conjecture for rooted trees. We also discuss the original conjecture for locally finite trees and state some new conjectures

Homogeneity in the free group

We show that any non abelian free group \F is strongly $\aleph_0$-homogeneous, i.e. that finite tuples of elements which satisfy the same first-order properties are in the same orbit under \Aut(\F). We give a characterization of elements in finitely generated groups which have the same first-order properties as a primitive element of the free group. We deduce as a consequence that most hyperbolic surface groups are not $\aleph_0$-homogeneous.Comment: 26 page