With one track, $\mathrm{TrackBad}(n,m,0)\iff H(n)>m$ (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. For $n\ge1$ and every $m$,

So the $s=0$ case of the track condition is exactly the statement that no map is $m$-good; together with not_trackBad_of_H_le, the track reformulation loses no strength and gains none.

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