Erdős Problem #624

The condition that an integer `m` ensures the existence of a function `f` covering `Fin n` for all large enough subsets `Y`. The property is invariant under bijection, so we use a representative `Fin n` for a finite set of size `n`.

Let \(H(n)\) be the minimum integer \(m\) such that there is a function \(f: \mathcal{P}(X) \to X\) where \(X\) is a finite set of size \(n\), such that for every subset \(Y \subseteq X\) with \(|Y| \ge m\), the set \(\{f(A) : A \subseteq Y\}\) covers \(X\).