# Logic Seminar: Infinite Combinatorics from Finite structures, by John Baldwin

April 3, 2024

4:00 PM - 4:50 PM

John Baldwin (UIC): Infinite Combinatorics from Finite structures

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.

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## Date posted

Apr 15, 2024

## Date updated

Apr 15, 2024