Recovering metric from full ordinal information

Given a geodesic space (E, d), we show that full ordinal knowledge on the metric d-i.e. knowledge of the function D d : (w, x, y, z) $\rightarrow$ 1 d(w,x)$\le$d(y,z) , determines uniquely-up to a constant factor-the metric d. For a subspace En of n points of E, converging in Hausdorff distance to E, we construct a metric dn on En, based only on the knowledge of D d on En and establish a sharp upper bound of the Gromov-Hausdorff distance between (En, dn) and (E, d)

Excellent Abstract Elementary Classes are tame

The assumption that an AEC is tame is a powerful assumption permitting development of stability theory for AECs with the amalgamation property. Lately several upward categoricity theorems were discovered where tameness replaces strong set-theoretic assumptions. We present in this article two sufficient conditions for tameness, both in form of strong amalgamation properties that occur in nature. One of them was used recently to prove that several Hrushovski classes are tame. This is done by introducing the property of weak $(\mu,n)$-uniqueness which makes sense for all AECs (unlike Shelah's original property) and derive it from the assumption that weak (\LS(\K),n)-uniqueness, (\LS(\K),n)-symmetry and (\LS(\K),n)-existence properties hold for all $n<\omega$. The conjunction of these three properties we call \emph{excellence}, unlike \cite{Sh 87b} we do not require the very strong (\LS(\K),n)-uniqueness, nor we assume that the members of \K are atomic models of a countable first order theory. We also work in a more general context than Shelah's good frames.Comment: 26 page

Limit Models in Strictly Stable Abstract Elementary Classes

In this paper, we examine the locality condition for non-splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove: Suppose that $K$ is an abstract elementary class satisfying 1. the joint embedding and amalgamation properties with no maximal model of cardinality $\mu$. 2. stabilty in $\mu$. 3. $\kappa_{\mu}(K)<\mu^+$. 4. continuity for non-$\mu$-splitting (i.e. if $p\in gS(M)$ and $M$ is a limit model witnessed by $\langle M_i\mid i<\alpha\rangle$ for some limit ordinal $\alpha<\mu^+$ and there exists $N$ so that $p\restriction M_i$ does not $\mu$-split over $N$ for all $i<\alpha$, then $p$ does not $\mu$-split over $N$). For $\theta$ and $\delta$ limit ordinals $<\mu^+$ both with cofinality $\geq \kappa_{\mu}(K)$, if $K$ satisfies symmetry for non-$\mu$-splitting (or just $(\mu,\delta)$-symmetry), then, for any $M_1$ and $M_2$ that are $(\mu,\theta)$ and $(\mu,\delta)$-limit models over $M_0$, respectively, we have that $M_1$ and $M_2$ are isomorphic over $M_0$.Comment: This article generalizes some results from arXiv:1507.0199