We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.

A field K in a ring language $\mathcal {L}$ is finitely undecidable if $\mbox {Cons}(T)$ is undecidable for every nonempty finite $T \subseteq {\mathtt{Th}}(K; \mathcal {L})$. We extend a construction of Ziegler and (among other results) use a first-order classification of Anscombe and Jahnke to prove every NIP henselian nontrivially valued field is finitely undecidable. We conclude (assuming the NIP Fields Conjecture) that every NIP field is finitely undecidable. This work is drawn from the author’s PhD thesis [48, Chapter 3].

Building on Pierre Simon’s notion of distality, we introduce distality rank as a property of first-order theories and give examples for each rankmsuch that$1\leq m \leq \omega $. For NIP theories, we show that distality rank is invariant under base change. We also define a generalization of type orthogonality calledm-determinacy and show that theories of distality rankmrequire certain products to bem-determined. Furthermore, for NIP theories, this behavior characterizesm-distality. If we narrow the scope to stable theories, we observe thatm-distality can be (...) characterized by the maximum cycle size found in the forking “geometry,” so it coincides with$(m-1)$-triviality. On a broader scale, we see thatm-distality is a strengthening of Saharon Shelah’s notion ofm-dependence. (shrink)

We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th where Pr is a predicate for the prime numbers and their negations is decidable, unstable, and supersimple. This is in contrast with Th which is known to be undecidable by the works of Jockusch, Bateman, and Woods.

We consider the structures $$, $$, $$, and $$ where $\mathbb {Z}$ is the additive group of integers, $\mathrm {SF}^{\mathbb {Z}}$ is the set of $a \in \mathbb {Z}$ such that $v_{p} < 2$ for every prime p and corresponding p-adic valuation $v_{p}$, $\mathbb {Q}$ and $\mathrm {SF}^{\mathbb {Q}}$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are (...) model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences. (shrink)