Tue, 20 Feb 2024, 1:25 pm MT
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 strongly minimal Steiner $k$-Steiner system $(M,R)$ from
[3] can be `coordinatized' in the sense of [4] by
a quasigroup if $k$ is a prime-power. But for the basic construction this
coordinatization is never definable in $(M,R)$ [1].
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.
We compare these structures with Steiner systems given by Fraïssé
constructions [2] and compare versions of the small
intersection property for $\aleph_0$-categorical structures and countably
saturated structures [5].
[1] John T. Baldwin.
Strongly minimal Steiner Systems II: Coordinatizaton and Strongly
Minimal Quasigroups.
Algebra Universalis, 84, 2023.
[2] Silvia Barbina and Enrique Casanovas.
Model theory of Steiner triple systems.
Journal of Mathematical Logic, 20, 2019.
[3] John T. Baldwin and G.~Paolini.
Strongly Minimal Steiner Systems I.
Journal of Symbolic Logic, 86:1486--1507, 2021.
[4] Bernhard Ganter and Heinrich Werner.
Equational classes of Steiner systems.
Algebra Universalis, 5:125--140, 1975.
[5] D. Lascar.
Automorphism groups of saturated structures; a review.
In Proceedings ICM 2002, vol 3, pages 25--33. 2002.