Strongly minimal Steiner Systems III: Path graphs and sparse configurations

Abstract

We introduce a uniform method of proof for the following results. For {\em each} of the following conditions, there are $2^{{\aleph }_0}$ families of Steiner systems, satisfying that condition: i) Theorem~2.2.4: (extending \cite{Chicoetal}) each Steiner triple system is $\infty$-sparse and has a uniform but not perfect path graph; ii) (Theorem~5.4.2: (extending \cite{CameronWebb}) each Steiner $k$-system (for $k=p^n$) is $2$-transitive and has a uniform path graph (infinite cycles only); iii) Theorem~2.1.5: (extending \cite{Fujiwaramitre}, each is anti-Pasch (anti-mitre); iv) Theorem~3.6 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.

27 pages, 2 figures The paper has been submitted to a combinatorics journal

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• Towards a Finer Classification of Strongly Minimal Sets