John Baldwin : Strongly minimal Steiner systems

Posted by James Freitag , part of the Logic Seminar.

At

Aug. 28, 2018, 3:30 p.m.

In

427 SEO

Abstract

With Gianluca Paolini, we constructed families of strongly minimal Steiner $(\infty,2,k)$ systems for every $k \geq 3$. Here we show that the $2^{\aleph_0}$ Steiner $(2,3)$-systems are coordinatized by strongly minimal Steiner quasigroups and the Steiner $(2,4)$-systems are coordinatized by strongly minimal $SQS$-Skeins. Further the Steiner $(2,4)$-systems admit Steiner quasigroups but it is open whether their theory is strongly minimal. We exhibit strongly minimal uniform Steiner triple systems (with respect to the associated graphs $G(a,b)$ (Cameron and Webb) with varying numbers of finite cycles. This work inaugurates a program of differentiating the many strongly minimal sets, whose geometries of algebraically closed sets are (locally isomorphic) to the original Hrushovski example, but with varying properties in the object language.