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We introduce a uniform method of proof for the following results. For {\emeach} of the following conditions, there are $2^{\aleph_0}$ families of Steinersystems, satisfying that condition: i) Theorem~2.2.4: (extending\cite{Chicoetal}) each Steiner triple system is $\infty$-sparse and has auniform but not perfect path graph; ii) (Theorem~5.4.2: (extending\cite{CameronWebb}) each Steiner $k$-system (for $k=p^n$) is $2$-transitive andhas 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 anexplicit quasi-group structure. In each case all members of the family satisfythe same complete strongly minimal theory and it has $\aleph_0$ countablemodels and one model of each uncountable cardinal.

Strongly minimal Steiner Systems III: Path graphs and sparse configurations