Logic Seminar
John Baldwin
UIC
Strongly minimal Steiner systems: combinatorics and classification
Abstract: Combinatorics:
We introduce a uniform method
of proof for the following results. For each of the following
conditions, there are $2^{\aleph_0}$ families of Steiner systems, satisfying
that condition: i) Theorem 1: (extending Chicot et al) each Steiner triple
system is $\infty$-sparse and has a uniform but not perfect path graph; ii)
(Theorem 2: (extending Cameron-Webb) each Steiner $k$-system (for $k=p^n$) is
$2$-transitive and has a uniform path graph (infinite cycles only); iii)
Theorem 3: (extending Fujiwara), each is anti-Pasch (anti-mitre); iv) Theorem
4 Steiner $k$-system has an explicit quasi-group structure. In each case all
members of the family satisfy the same complete strongly minimal theory and
it has $\aleph_0$ countable models and one model of each uncountable
cardinal. Item iii) is particularly interesting since it fails completely
for the most obvious construction of such Steiner systems, which fall in 1)
of the classification below.
Classification: We will briefly discuss the lengthy proof (with Verbovskiy) that the strongly minimal Steiner systems and Hrushovski's
original example a) do not admit a) elimination of imaginaries or (more
strongly) b) an $\emptyset$-definable binary function. This result justifies
the observation that changing the $\mu$-function or adding axioms like linear
space yields profoundly different strongly minimal sets. The ab initio
Hrushovski construction yields
non-trivial flat classes which split into those
i) with
no definable binary function, ii)
definable binary functions exist. These can be further subdivided as: a) no
commutative binary function (elimination of imaginaries fails); b) strongly
minimal quasigroups discussed in the first part of the
talk;
c) non-Desarguesian projective planes coordinatized by ternary fields.
Tuesday April 26, 2022 at 4:00 PM in 636 SEO