STRONGLY MINIMAL STEINER SYSTEMS I: EXISTENCE

Abstract

AbstractA linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner k-system (for $k \geq 2$ ) 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 (bi-interpretable) vocabulary $\tau $ with a single ternary relation R. We prove that for every integer k there exist $2^{\aleph _0}$ -many integer valued functions $\mu $ such that each $\mu $ determines a distinct strongly minimal Steiner k-system $\mathcal {G}_\mu $ , whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture.