Random obstruction: $\mathrm{TrackBad}(n,m,s)$ fails once $2^{m+s}\ge n(\ln n\,(m+1)+1)$ (Erdős 624)

Let $H(n)$ be the least $m$ such that some $f:\mathcal P(X)\to X$, $|X|=n$, satisfies $\{f(A):A\subseteq Y\}=X$ for all $Y\subseteq X$ with $|Y|\ge m$ (Erdős Problem 624). Say $\mathrm{TrackBad}(n,m,s)$ holds if for every family of maps $g_B:\mathcal P(U)\to U$ ($|U|=n$) indexed by the subsets $B$ of an $s$-set, there are $Y\subseteq U$ with $|Y|=m$ and $u\in U$ with $u\notin\{g_B(A):A\subseteq Y\}$ for every $B$. By an earlier result, $\mathrm{TrackBad}(n,m,s)$ implies $H(n+s)>m+s$ for $n\ge1$. Theorem. If $n\ge2$ and

then $\mathrm{TrackBad}(n,m,s)$ is false. In particular the track method can never give more than $H(n+s)>\log_2 n+2\log_2\log_2 n+O(1)$.

From $\binom nm\le n^m$ and $(1-1/n)^K\le e^{-K/n}$ one gets $n\binom nm(n-1)^K\le e^{-1}n^K<n^K$ for $K=2^{m+s}$; then apply the counting obstruction.

Formalization Note. $U=\mathrm{Fin}\,n$ and the $s$-set is $\mathrm{Fin}\,s$, as in Erdos624.track_amplification.