No 17‐player triplewhist tournament has nontrivial automorphisms

Abstract

The existence of triplewhist tournaments for v players has recently been solved for all values of v except v = 17. For v = 12 and v = 13 a complete enumeration has shown the nonexistence of T W h (v), while constructions of T W h (v) have been presented for v > 17. For several values of v existence has been shown by constructing a T W h (v) with a prescribed, usually cyclic, automorphism group. In this article it is shown that the strategy of constructing a T W h (v) with a prescribed automorphism group cannot succeed with T W h (17), since no 17-player triplewhist tournament has nontrivial automorphisms.

New Z-cyclic triplewhist frames and triplewhist tournament designs

“…Since the planes of PG (3,5) intersect in lines of PG (3,5), we can look at the intersection of shift 0 with shift i. Successively taking i to be differences not occurring in the intersections so far gathered, we see we take shifts of i = 1, 2, 3, 4, 17, yielding the intersections: (1,2,8,17,28), (2,4,16,34,56), (4,7,17,46,54), (4,8,27,32,68), (0, 17, 34, 51, 68).…”

“…The initial round of a Z-cyclic TWhFrame(7 5 ) is given by the following seven games. (13,21,14,27), (17,18,26,29), (9,16,12,33), (32, 3,1,19), (34,8,7,31), (4,2,11,23), (22,6,24,28).…”

“…, 8. Applying the same halving to row 0 of the log table produces the two base blocks (∞, 0, 9,27) and (18,54,36,45) and adjoining these two blocks to the initial frame round gives a TWh(64).…”

“…Although it is not known whether or not a TWh(17) exists, it is known that a Z-cyclic TWh(17) does not exist [17]-it is even known that if a TWh (17) were to exist, its only automorphism would be the trivial one [27]. Thus it is natural to question whether or not Z-cyclic TWh(5 c 13 d 17 e ) exist for some combination of the integers c, d, e. That there are affirmative answers to this question is the focus of this section.…”

“…For a Z-cyclic WhFrame (15 17 ), consider the difference set for PG (3,5). The (85, 21, 5) BIBD is given by the cyclic difference set (0, 1,2,4,7,8,14,16,17,23,27,28,32,34,43,46,51,54,56,64,68).…”