STRONGLY MINIMAL STEINER SYSTEMS I: EXISTENCE

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 (bi-interpretable) vocabulary z with a single ternary relation R. We prove that for every integer k there exist 2.1 -many integer valued functions g such that each g determines a distinct strongly minimal Steiner k-system ci, whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture.