Towards a Characterisation of Lottery Set Overlapping Structures

Consider a lottery scheme consisting of randomly selecting a winning t–set from a universal m–set, while a player participates in the scheme by purchasing a playing set of any number of n–sets from the universal set prior to the draw, and is awarded a prize if k or more elements of the winning t–set occur in at least one of the player’s n–sets (1 ≤ k ≤ {n, t} ≤ m). This is called a k–prize. The player may wish to construct a playing set, called a lottery set, which guarantees the player a k–prize, no matter which winning t–set is chosen from the universal set. The cardinality of a smallest lottery set is called the lottery number, denoted by L(m, n, t; k), and the number of such non–isomorphic sets is called the lottery characterisation number, denoted by η(m, n, t; k). In this paper an exhaustive search technique is employed to characterise minimal lottery sets of cardinality not exceeding six, within the ranges 2 ≤ k ≤ 4, k ≤ t ≤ 11, k ≤ n ≤ 12 and max{n, t} ≤ m ≤ 20. In the process 32 new lottery numbers are found, and bounds on a further 31 lottery numbers are improved. We also provide a theorem that characterises when a minimal lottery set has cardinality two or three. Values for the lottery characterisation number are also derived theoretically for minimal lottery sets of cardinality not exceeding three, as well as a number of growth and decomposition properties for larger lotteries.