On sparse countably infinite Steiner triple systems

Abstract

Abstract:We give a general construction for Steiner triple systems on a countably infinite point set and show that it yields 2 ℵ 0 nonisomorphic systems all of which are uniform and r-sparse for all finite r ≥ 4. q

Strongly minimal Steiner Systems III: Path graphs and sparse configurations

“…The notion of an ∞-sparse system uniformizes these anti-x constructions [Fuj06b,CGGW10]. We derive such results for infinite Steiner triple systems by variants on our general construction.…”

“…Note that the Pasch, mitre, and mia configurations are all forbidden in an ∞sparse STS. [CGGW10] construct by a four page inductive construction of finite approximations, 2 ℵ0 non-isomorphic countable ∞-sparse systems. We modify the construction in [BP20] by restricting K 0 to K sp 0 to get ∞-sparse STS of every infinite cardinality.…”

“…In Section 5.2 we introduce the notion of a uniform (The isomorphism type of G M (a, b) does not depend on the choice of a, b.) q-Steiner system (generalizing [CW12,CGGW10]). Then by different methods in Sections 5.3 and 5.4 we construct families of 2-transitive and hence uniform q-Steiner systems.…”

“…Section 1 sketches background information on the Hrushovski construction and its relation to other generalization of the Fraïssé method. We reformulate (Section 2) the Cameron-Webb notion of sparse configurations [CGGW10,Fuj06a] in terms of the δ-function fundamental to the Hrushovski construction and give uniform accounts of the existence in every infinite cardinality of anti-Pasch, anti-mitre and indeed ∞-sparse Steiner triple systems (Corollaries 2.1.4 and 2.1.5 and Theorem 2.2.4). While the examples of strongly minimal pure Steiner k-systems, (M, R), with k > 3 admit no definable 'truly binary' operation with infinite domain [BV21,Bal21], we construct (Theorem 3.6) strongly minimal quasigroups which induce q-Steiner systems (line length q) for q a prime power.…”

Sparse Steiner triple systems of order 21

“…Erdős's conjecture is still open for this case and is therefore open for all . Families of 5‐sparse [18] and 6‐sparse [8] Steiner triple systems have been discovered, but not a single nontrivial example is known for (see, however, [1] for infinite systems).…”