Triplewhist tournaments with the three person property

The necessary conditions for existence of a triplewhist tournament TWh(v) are . By the efforts of many authors through a century, these conditions are shown to be sufficient except for v=5,9,12,13 and possibly for v=17. A triplewhist tournament Wh(v) is said to have the three person property if any two games in the tournament do not have three common players. We briefly denote such a design as a 3PTWh(v). In this paper, we extend the known existence result for TWh(v)s and show that the necessary conditions for existence of a 3PTWh(v), namely, v⩾8 and , are also sufficient except for v=9,12,13 and possibly for v=17.