Establishing a property of finite sets (Q2730585)

For all \(k\in \mathbb N \) let us consider \( n(k)\in \mathbb N \) such that \( 2^{n(k)-1}<k+2\leq 2^{n(k)}\). The author gives minimal values of \(s\) for the following problem: let \( A_1,\ldots ,A_k\) be subsets of a finite set \(X\) whose cardinalities are greater than a half of the cardinality of \(X\). Then we can choose \(s\) elements of \(X\) in such a way that any \(A_i\) includes at least one element from these \(s\) elements.