Simultaneous Systems of Representatives for Families of Finite Sets

Melvyn B. Nathanson
1988 Proceedings of the American Mathematical Society  
Let h > 2 and k > 1. Then there exists a real number A = \(h,k) e (0,1) such that, if S" = {Si}si=1 and 3~ = {3>}i_, are families of nonempty, pairwise disjoint sets with |S¿| < h and |Tj-| < k and S¿ C¿ Tj for all i and j, then N(5^,^) < /i3A(, where N(5^,J') is the number of sets X such that X is a minimal system of representatives for 5? and X is simultaneously a system of representatives for !?. A conjecture about the best possible value of the constant \(h, k) is proved in the case h > k.
more » ... he necessity of the disjointness conditions for the families J/" and if is also demonstrated. Let 5? = {Si}¡=1 and ET = {T}J=1 be two families of nonempty sets. The set A is a system of representatives for S? if X f~l S% ^ 0 for all i G I. If A is a system of representatives for 5? but no proper subset of A is a system of representatives for S", then A is a minimal system of representatives for S?. Let M'S?) denote the number of minimal systems of representatives for S*. If the sets S% are pairwise disjoint, and if |5¿| < h, where |S| denotes the cardinality of the set S, then M(S") = V[si=1 |5¿| < hs.
doi:10.2307/2047133 fatcat:wgwvcltvwbhalicw65vzqlu6m4