Systems of Representatives

Lower and upper bounds are obtained for the size $\zeta(n,r,s,k)$ of a minimum system of common representatives for a system of families of $k$-element sets. By $\zeta(n,r,s,k)$ we mean the maximum (over all systems $\Sigma=\{M_1,\dots,M_r\}$ of sets $M_i$ consisting of at least $s$ subsets of $\{1,\dots,n\}$ of cardinality not exceeding $k$) of the minimum size of a system of common representatives of $\Sigma$. The obtained results generalize previous estimates of $\zeta(n,r,s,1)$.