Related papers: More on tree properties

We introduce the idea of a weakly entangled linear order, and show that it is consistent for a Suslin line to be weakly entangled. We generalize the notion of entangled linear orders to $\omega_1$-trees, and prove that an $\omega_1$-tree is…

We observe that the definition of Shelah's classical $\mathrm{NSOP}_{n}$ hierarchy for first-order theories, for integers $n \geq 3$, can be restated so that it extends to the case where $n$ is replaced with any real number $r \geq 3$.…

Tangle-tree theorems are an important tool in structural graph theory, and abstract separation systems are a very general setting in which tangle-tree theorems can still be formulated and proven. For infinite abstract separation systems, so…

We present in this paper a first-order axiomatization of an extended theory $T$ of finite or infinite trees, built on a signature containing an infinite set of function symbols and a relation $\fini(t)$ which enables to distinguish between…

A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if a countable theory T has the Schroder-Bernstein property then it is classifiable (it is…

We investigate a model theoretic invariant $\kappa_{srd}^m(T)$, which was introduced by Shelah in his famous book, and prove that $\kappa_{srd}^m(T)$ is sub-additive. When $\kappa_{srd}^m(T)$ is infinite, this gives the equality…

We show that the existence of a well-known type of ideals on a regular cardinal $\lambda$ implies a compactness property concerning the specialisability of a tree of height $\lambda$ with no cofinal branches. We also use Neeman's method of…

Sturm's oscillation theorem states that the n-th eigenfunction of a Sturm-Liouville operator on the interval has n-1 zeros (nodes). This result was generalized for all metric tree graphs and an analogous theorem was proven for discrete tree…

We consider existentially closed fields with several orderings, valuations, and $p$-valuations. We show that these structures are NTP$_2$ of finite burden, but usually have the independence property. Moreover, forking agrees with dividing,…

I develop in depth the machinery of $(\mathcal L, n)$-models originally introduced by Shelah and, independently in a slightly different form by Kripke. This machinery allows fairly routine constructions of true but unprovable sentences in…

A class of structures is monadically dependent if one cannot interpret all graphs in colored expansions from the class using a fixed first-order formula. A tree-ordered $\sigma$-structure is the expansion of a $\sigma$-structure with a…

We streamline treatments of the interpretability orders $\trianglelefteq^*_\kappa$ of Shelah, the key new notion being that of pseudosaturation. Extending work of Malliaris and Shelah, we classify the interpretability orders on the stable…

A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; the core companion of a theory $T$ is a model-complete core theory $S$ such that every model…

We investigate the generalized tree properties and guessing model properties introduced by Wei\ss\ and Viale, as well as natural weakenings thereof, studying the relationships among these properties and between these properties and other…

We present a new manifestation of G\"odel's second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert's program. Specifically, we consider a proper extension of Peano arithmetic…

A $k$-ended tree is a tree with at most $k$ leaves. In this note, we give a simple proof for the following theorem. Let $G$ be a connected graph and $k$ be an integer ($k\geq 2$). Let $S$ be a vertex subset of $G$ such that $\alpha_{G}(S)…

In this paper, we introduce two families of planar and self-similar graphs which have small-world properties. The constructed models are based on an iterative process where each step of a certain formulation of modules results in a final…

Let F be a set of relational trees and let Forbh(F) be the class of all structures that admit no homomorphism from any tree in F; all this happens over a fixed finite relational signature $\sigma$. There is a natural way to expand Forbh(F)…

We provide a model-theoretic classification of the countable homogeneous $\mathbf{H}_4$-free 3-hypertournament studied by Cherlin, Hubi\v{c}ka, Kone\v{c}n\'y, and Ne\v{s}et\v{r}il. Our main result is that the theory of this structure is…

In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for $\mu$-forking, a certain independence notion arising from splitting. This had been a longstanding technical…