Hyperovale in Steiner-Tripel-Systemen (Q1059857)

By analogy with finite Desarguesian planes, the authors investigate ovals and hyperovals in a Steiner triple system S. A subset Q is called an oval if through each of its points there is exactly one block, called the tangent, containing no other point of Q. If the tangents to an oval have a point K in common, then \(Q\cup \{K\}\) is called a hyperoval. A Steiner triple system on v points containing a hyperoval is denoted by HSTS(v). It is shown that an HSTS(v) exists if and only if \(v>3\) and \(v\equiv 3\) or 7 (mod 12). This is done by constructing an \(HSTS(1+2v)\) from an STS(v). Two such constructions are given which result in non-isomorphic \(HSTS(1+2v)'s\).