Strongly Minimal Steiner Systems I: Existence

Abstract

A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner k-system (for k 2) is a linear space such that each line has size exactly k. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a (biinterpretable) vocabulary τ with a single ternary relation R. We prove that for every integer k there exist 2 ℵ 0 -many integer valued functions µ such that each µ determines a distinct strongly minimal Steiner k-system Gµ, whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture.

Strongly minimal Steiner Systems III: Path graphs and sparse configurations

“…The associative law forces the intrinsic closure icl(P) of three algebraically independent elements (which should have d(icl(P) 3) to have dimension 2. Nevertheless, in general there will be many realizations of a Pasch configuration P in a strongly minimal Steiner triple system constructed as in [BP20], since δ(P ) = 2 0. Indeed any pair of points extends to a Pasch configuration in the generic model.…”

“…[CGGW10] construct by a four page inductive construction of finite approximations, 2 ℵ0 non-isomorphic countable ∞-sparse systems. We modify the construction in [BP20] by restricting K 0 to K sp 0 to get ∞-sparse STS of every infinite cardinality. Definition 2.2.2.…”

“…We use the first clause of # to avoid B with δ(B) = 1. Now the proof from [BP20] applies to give amalgamation for K µ if µ ∈ U sp .…”

“…However, there are ternary fields, Steiner systems and quasigroups are among them. [Bal94,BP20,Bal21,BV21] shows that structural distinctions arise by fixing a class U of permissible choices for the function µ. In fact, the family of 'Hrushovski constructions of strongly minimal sets' depend on five parameters.…”

“…A linear space is collection of points and lines such that two points determine a line, a minimal condition to call a structure a geometry. A linear space is a Steiner k-system if every line (block) has cardinality k. We showed in Section 2 of [BP20] that linear spaces can be naturally formulated in a one-sorted logic with single ternary 'collinearity' predicate and proved the existence of strongly minimal Steiner q-systems for every prime power. These theories are model complete and satisfy the usual properties of counterexamples to Zilber's trichotomy conjecture: Their acl-geometries are non-trivial, not locally modular, and the theory cannot interpret an infinite group.…”