Spanning sets and scattering sets in Steiner triple systems (Q1176376)

Given a Steiner triple system of \((V,{\mathcal B})\) a set \(X\subseteq V\) is a spanning set if for every \(v\in V\backslash X\) there is a \(B\in{\mathcal B}\) so that \(v\in B\) and \(| B\cap X|=2\). The set \(X\) is a scattering set if for every \(B\in{\mathcal B}\), \(| X\cap B|\leq 2\) and for every \(y\in V\backslash X\) there is at most one \(B\in{\mathcal B}\) so that \(y\in B\) and \(| B\cap X|=2\). For each \(v\equiv 1\), \(3\pmod 6\) the authors construct a Steiner triple system with a spanning set of minimum cardinality \(\left(\left\lceil{1\over 2}(\sqrt{8v+1}- 1)\right\rceil\right)\) and another with a scattering set of maximum cardinality \(\left(\left\lfloor{1\over 2}(\sqrt{8v+1}- 1)\right\rfloor\right)\). As a consequence, the size of minimum cardinality complete arcs in a Steiner triple system of order \(v\) is determined.