Hyperovals in Steiner triple systems
A hyperoval in a Steiner triple system is the complement of a subsystem of maximum possible size, i.e. a hyperoval is a set of points met by each block in either two or zero points. For every \(v\equiv 1\) or \(3\pmod 6\) the authors define: \[ \text{HSTS}(v)=\bigl\{ h\in N:\;\text{there is an } \text{STS}(v)\text{ containing \(h\) hyperovals}\bigr\}\quad\text{and} \] \[ \text{H}(v)=\bigl\{2^ n-1: n\in N\text{ and } (v-2^ n+1)/2^ n\equiv 1\text{ or }3\pmod 6\bigr\}, \] and they show that \(\text{HSTS}(v)=\text{H}(v)\) for \(v>7\). The proof is by construction, relies on well-known results on Steiner triple systems, and uses standard techniques, like 1-factorizations of suitable complete graphs.
scientific article; zbMATH DE number 428602