Triplewhist tournaments with the three person property

Abstract

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 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 v ≡ 0, 1 (mod 4), are also sufficient except for v = 9, 12, 13 and possibly for v = 17.

Ordered tournaments and ordered triplewhist tournaments with the three person property

“…The initial round of games is obtained by multiplying the block (x, x 2 , x 31 = 2, x 84 = 4x + 4) by x 4t for 0 ≤ t ≤ 30. (18,10,19,26), (7,16,31,12), (21,20,27,9), (15,25,28,11), (1,13,29,23), (24,22,3,14), (30,17,4,32). 45 (1,3,5,21), (4,43,6,34), (33,41,14,29), (7,25,32,10), (44,12,31,28), (13,17,8,18), (22,<...>…”

“…Concretely, we have the following theorem. Theorem 1.3 [24]. Necessary conditions for existence of a 3PTWh(v) are v ≡ 0, 1 (mod 4) and v ≥ 8.…”

“…As more and more results have been obtained, the attention has turned to whist tournaments that satisfy more than one of the above mentioned criteria simultaneously (see, e.g. [1,8,11,13,23,24,28,32]). In what follows, for convenience, we shall provide a brief description of such types of tournaments and the known results associated with them.…”

“…Recently Ge [24] gave an almost solution for the existence of TWh(v)s with the three person property (briefly denoted by 3PTWh(v)). Concretely, we have the following theorem.…”