Complete arcs in Steiner triple systems (Q1374194)

A complete arc in a Steiner triple system of order \(v\), \(\text{STS}(v)\), is a set of \(s\) points met by any block in at most two points and such that the 2-secant blocks cover all points outside the arc. The authors determine the spectrum of sizes of complete arcs in Steiner triple systems; namely, they prove the following theorem: If \(v\equiv 1, 3\pmod 6\) and \(\lceil(\sqrt{8v+ 1}-1)/2\rceil\leq s\leq T(v)\) (where \(T(v)\equiv(v+1)/2\) when \(v\equiv 3, 7\pmod{12}\) and \(T(v)\equiv(v- 1)/2\) when \(v\equiv 1, 9\pmod{12})\), then there exists an \(\text{STS}(v)\) containing a complete \(s\)-arc. When \(v=7\) or \(v\geq 15\), this \(\text{STS}(v)\) has a subsystem \(\text{STS}(7)\) which contains at least two points of the complete \(s\)-arc. When \(v\geq 15\), for the complete arc and sub-\(\text{STS}(7)\) chosen, there is a 2-secant block of the arc which meets the sub-\(\text{STS}(7)\) in one point off the arc. The proof is by constructions for the possible cases. One of the main tools is the singular direct product. Other standard constructions, as the \(v\to 2v+1\), are employed. Several examples for small values of \(v\) are provided.