\(\mathbb Z\)CPS-starters, a necessary and sufficient condition for the existence of \(\mathbb Z\)CPS whist designs (Q2799691)

A whist tournament, Wh\((v)\) on \(v\) players is a schedule of games each involving two players against two others, such that each player partners every other player exactly once and each player opposes every other player exactly twice. For \(v=4n\), the games are arranged in \(4n-1\) rounds, each of \(n\) games, and each player plays in exactly one game in each round. For \(v=4n+1\), the games are arranged in \(4n+1\) rounds, each of \(n\) games, and each player plays in exactly one game in all but one of the rounds. It is well known that Wh\((v)\) exist for all \(v\equiv 0,1 \pmod{4}\). Note that a Wh\((v)\) is a resolvable (near resolvable) \((v,4,3)\)-BIBD where each block is considered to be a whist game or table. A whist tournament is called \(Z\)-cyclic if the players are the elements in \(Z_{4n-1} \cup \{\infty\}\) or \(Z_{4n+1}\) (resp.) and the rounds are cyclic.NEWLINENEWLINELet \(G\) be an abelian group of order \(2k+1\) and let \(e_{G}\) denote the identity element. The collection of pairs \(\{\{x,-x\} : x \in G, x\neq e_{G}\}\) is called the patterned starter. In this note, the author shows that patterned starters can be used to construct a set of initial round partner pairs for \(Z\)-cyclic whist tournaments; these are called \(ZCPS\) starters and the resulting designs are called \(ZCPS\) whist tournaments. The main result of this paper is to establish that the existence of a \(ZCPS\)-starter is a necessary and sufficient condition for the existence of a \(ZCPS\) whist tournament.