Triplewhist tournaments with the three person property

The necessary conditions for existence of a triplewhist tournament TWh ( v ) are v ≡ 0 , 1 ( mod 4 ) . 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 3 PTWh ( v ) . In this paper, we extend the known existence result for TWh ( v ) s and show that the necessary conditions for existence of a 3 PTWh ( v ) , namely, v ⩾ 8 and v ≡ 0 , 1 ( mod 4 ) , are also sufficient except for v = 9 , 12 , 13 and possibly for v = 17 .