arXiv subjects
Publications and source records attributed to Omer Mermelstein.
Denote Hrushovski's non-collapsed ab initio construction for an $n$-ary relation by $\mathcal{M}_{\not\sim}$ and the analogous construction for a symmetric $n$-ary relation by $\mathcal{M}_{\sim}$. We show that $\mathcal{M}_{\not\sim}$ is isomorphic to a proper reduct of $\mathcal{M}_{\sim}$ and vice versa, and that the combinatorial pregeometries associated with both structures are isomorphic.
We build a new spectrum of recursive models (SRM(T)) of a strongly minimal theory. This theory is non-disintegrated, flat, model complete, and in a language with a finite signature.
We show that for a model complete strongly minimal theory whose pregeometry is flat, the recursive spectrum (SRM($T$)) is either of the form $[0,α)$ for $α\in ω+2$ or $[0,n]\cup\{ω\}$ for $n\in ω$, or $\{ω\}$, or contained in $\{0,1,2\}$. Combined with previous results, this leaves precisely 4 sets for which it is not yet determined whether each is the spectrum of a model complete strongly minimal theory with a flat pregeometry.
We examine the first order structure of pregeometries of structures built via Hrushovski constructions. In particular, we show that the class of flat pregeometries is an amalgamation class such that the pregeometry of the unbounded arity Hrushovski construction is precisely its generic. We show that the generic is saturated, provide an axiomatization for its theory, show that the theory is $ω$-stable, and has quantifier-elimination down to boolean combinations of $\exists\forall$-formulas. We show that the pregeometries of the bounded-arity Hrushovski constructions satisfy the same theory, and that they in fact form an elementary chain.
Closed ordinal Ramsey numbers are a topological variant of the classical (ordinal) Ramsey numbers. We compute the exact value of the closed ordinal Ramsey number $R^{cl}(ω^2,3) = ω^6$.
We show that $R^{cl}(ω\cdot 2,3)^2$ is equal to $ω^3\cdot 2$.
Let $\mathbb{M}_n$ denote the structure obtained from Hrushovski's (non collapsed) construction with an n-ary relation and $PG(\mathbb{M}_n)$ its associated pre-geometry. It was shown by Evans and Ferreira that $PG(\mathbb{M}_3)\not\cong PG(\mathbb{M}_4)$. We show that $\mathbb{M}_3$ has a reduct, $\mathbb{M}^{clq}$ such that $PG(\mathbb{M}_4)\cong PG(\mathbb{M}^{clq})$. To achieve this we show that $\mathbb{M}^{clq}$ is a slightly generalised Fraïssé-Hrushovski limit incorporating into the construction non-eliminable imaginary sorts in $\mathbb{M}^{clq}$.
We show that the geometry of Hrushovski's ab initio construction for a single $n$-ary relation not-permitting dependent sets of size less than $n$, when restricted to $n$-tuples, can be itself constructed as a Hrushovski construction.
We show that the rank ω structure obtained by the non-collapsed version of Hrushovski's amalgamation construction has a proper reduct. We show that this reduct is the Fraïssé-Hrushovski limit of its own age with respect to a pre-dimension function generalising Hrushovski's pre-dimension function. It follows that this reduct has a unique regular type of rank ω, and we prove that its geometry is isomorphic to the geometry of the generic type in the original structure. We ask whether our reduct is bi-interpretable with the original structure and whether it, too, has proper reducts with the same geometry.