Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:\{A : A\subseteq X\}\to X$ so that for every $Y\subseteq X$ with $\lvert Y\rvert \geq H(n)$ we have

$$\{ f(A) : A\subseteq Y\}=X.$$

Prove that

$$H(n)-\log_2 n \to \infty.$$