Logic Seminar
John Baldwin
UIC
Infinite Combinatorics from Finite structures
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.
Wednesday April 3, 2024 at 4:00 PM in 712 SEO