An existence theorem for cyclic triplewhist tournaments
A triplewhist tournament on \(v= 4m+ 1\) players is a schedule of games \((a,b,c,d)\) involving two players \(a\) and \(c\) opposing two other players \(b\) and \(d\). The games are arranged into \(4m+ 1\) rounds such that each player plays in exactly one game in all but one round, partners each other player exactly once and opposes each other player exactly once in each of the possible two positions (as \(b\) or \(d\)-player, respectively). Such a tournament is called cyclic if the players are taken as the residues modulo \(v\) and the rounds are obtained from an initial round by developing modulo \(v\). The authors provide the first general existence theorem for this type of structure by proving the existence of a cyclic triplewhist tournament for \(v\) players whenever \(v\) is a product of powers of primes \(\equiv 5\pmod 8\). The interesting method of proof rests on three constructions together with a proof of the following result on primitive roots modulo \(p\) which is of independent interest: Given any prime \(p\equiv 5\pmod 8\) with \(p\geq 29\), \(p\neq 61\), there exists a primitive root \(\omega\text{ mod }p\) such that both \(\omega^ 2+ \omega+ 1\) and \(\omega^ 2- \omega+ 1\) are squares. Naturally enough, this result is proved by using character sums.
scientific article; zbMATH DE number 743994