OPEN

Let $X$ be a finite set of size $n$ and $H(n)$ be such that there is a function $f:P(X)\to X$ so that for every $Y\subseteq X$ with $\lvert Y\rvert \geq H(n)$ we have
\[\{ f(A) : A\subseteq Y\}=X.\]
Prove that
\[H(n)-\log_2 n \to \infty.\]

A problem of Erdős and Hajnal [ErHa68] who proved that
\[\log_2 n \leq H(n) < \log_2n +(3+o(1))\log_2\log n.\]
Erdős said that even the weaker statement that for $n=2^k$ we have $H(n)\geq k+1$ is open, but Alon has provided the following simple proof: by the pigeonhole principle there are $\frac{n-1}{2}$ subsets $A_i$ of size $2$ such that $f(A_i)$ is the same. Any set $Y$ of size $k$ containing at least $k/2$ of them can have at most
\[2^k-\lfloor k/2\rfloor+1< 2^k=n\]
distinct elements in the union of the images of $f(A)$ for $A\subseteq Y$.

For this weaker statement, Erdős and Gyárfás conjecture the stronger form that if $\lvert X\rvert=2^k$ then, for any $f:P(X)\to X$, there must exist some $Y\subset X$ of size $k$ such that \[\#\{ f(A) : A\subseteq Y\}< 2^k-k^C\] for every $C$ (with $k$ sufficiently large depending on $C$).