Some new triplewhist tournaments TWh(v)

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,<...>…”

“…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), (22,2,15,39), (40,9,19,26), (38,37,16,27), (35,30,24,36), (20,11,23,42). 49 (42, 40, 39, 24), (27,48,44,3), (23,45,10,16), (46,37,28,…”

“…49 (42, 40, 39, 24), (27,48,44,3), (23,45,10,16), (46,37,28,12), (8,26,31,18), (19,15,38,13), (43,33,21,32), (36,41,25,6), (20,34,5,22), (2, 1, 9, 29), (4,11,14,17), (7,30,47,35). 57 (25,14,1,56), (10,31,26,40), (2, 35, 52, 39), (9,47,19,11), (29,38,4,…”

“…For n = 7, the required design is over ( g Initial Games 8 (4, 21, 27, 13), (2,26,23,14), (1,7,3,14), (1,37,4,3), (2,29,28,16), (3,22,39,21), (1,8,19,17), (2, 39, 31, 23). 9 (2,36,23,14), (3,27,21,14), (3,22,1,29), (2, 43, 14, 1), (2,3,6,29), (4,22,18,6), (4,43,12,26), (1,9,37,8), (3,…”

“…They proved that the necessary condition for the existence of a TWh(v), namely v ≡ 0 or 1 (mod 4), is also sufficient with two definite exceptions, namely v = 5, 9, as well as 15 possible exceptions, namely v ∈ {12, 56} ∪ {13, 17, 45, 57, 65, 69, 77, 85, 93, 117, 129, 133, 153}. Subsequent improvements were made by Ge and Zhu in [29], Ge and Lam [27], and finally by Abel and Ge [5]. We summarize the known results for TWh(v) in the following theorem.…”