John Baldwin : Infinite Combinatorics from Finite structures

Posted by Matthew Harrison-Trainor , part of the Logic Seminar.

At

April 3, 2024, 4 p.m.

In

712 SEO

Abstract

We survey variants of the Hrushovki non-locally modular strongly minimal sets construction that get more combinatorial examples and show the basic construction is essentially unary and thus does not eliminate imaginaries. A $t-(\kappa,k,s)$ block design is a set of $\kappa$ elements and a collection of $k$-element subsets $B$ of $P$ (called blocks) with the property that each $t$-element subset of $P$ occurs in exactly $s$ blocks. A $k$-Steiner system is a $2-(\kappa,k,1)$ system. Using variants of the Hrushovski method we construct infinite block designs and Steiner systems that are a) $\aleph_1$-categorical and with more work b) have $t$-transitive automorphism groups for prescribed $t$. The high transivity is on-going work with Freitag and Mutchnik. The strongly minimal Steiner $k$-Steiner system $(M,R)$ from Baldwin and Paolini can be `coordinatized' in the sense of Ganter and Werner by a quasigroup if $k$ is a prime-power. But for the basic construction this coordinatization is never definable in $(M,R)$. In almost all cases the theory does not admit elimination of imaginaries. Nevertheless, by refining the construction, if $k$ is a prime power there is a $(2,k)$-variety of quasigroups which is strongly minimal and definably coordinatizes a Steiner $k$-system.